1066 ieee journal of quantum electronics, …sina.sharif.edu/~khorasani/publications/64.pdf · 1066...

1066 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 49, NO. 12, DECEMBER 2013

Simulation of Multipartite CavityQuantum Electrodynamics

Moslem Alidoosty, Sina Ataollah Khorasani, Senior Member, IEEE, and Mohammad Hasan Aram

Abstract— Cavity quantum electrodynamics of multipartitesystems are studied in depth, which consists of an arbitrarynumber of emitters in interaction with an arbitrary number ofcavity modes. The governing model is obtained by taking the fullfield-dipole and dipole-dipole interactions into account, and issolved in the Schrödinger picture with assumption of vanishingfield and dipole interactions at high energies. An extensive code isdeveloped that is able to solve the system and track its evolutionin time, while maintaining sufficient degrees of arbitrariness insetting up the initial conditions and interacting partitions. Usingthis code, we have been able to numerically evaluate variousparameters such as probabilities, expectation values (of field andatomic operators), and the concurrence as the most rigorouslydefined measure of entanglement of quantum systems. We presentand discuss several examples including a seven-partition systemconsisting of six quantum dots interacting with one cavity mode.We observe for the first time that the behavior of quantumsystems under ultrastrong coupling is significantly different thanthe weakly and strongly coupled systems, marked by onsetof chaos and abrupt phase changes. We also discuss how toimplement spin into the theoretical picture and thus, successfullysimulate a recently reported spin-entanglement experiment.

Index Terms— Cavity quantum electrodynamics, entanglement,quantum optics, spin.

I. INTRODUCTION

W ITHOUT doubt, Cavity Quantum Electrodynamics(CQED) is one of the frontiers of modern science,

where its applications are rapidly entering the realm of engi-neering fields. With the advent of quantum computing, CQEDremains at the cutting edge of the technology in this area,as the most successful and only commercialized platform.This is while the increasing number of quantum sub-systems,or the so-called partitions, demand for more complicatedanalytical and simulation tools capable of dealing with mul-tipartite systems without losing accuracy. As the number ofpartitions increase, the difficulty in treatment of multipartitesystems unfolds in two aspects: (a) how to maintain accuracywhile increasing the system dimensionality, and (b) how towrite down equations using proper notations and mathematicalexpression without causing confusion and/or ambiguity. This is

Manuscript received July 12, 2013; revised September 12, 2013 andSeptember 28, 2013; accepted October 15, 2013. Date of publicationOctober 21, 2013; date of current version October 30, 2013. This work wassupported by the Iranian National Science Foundation under Grant 89001329.

The authors are with the School of Electrical Engineering, Sharif Universityof Technology, P. O. Box 11365-9363, Tehran, Iran (e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JQE.2013.2286578

while we have to add the computational complexity of solvingalgorithm unaddressed.

The first successful theoretical understanding of CQEDwas made by Jaynes and Cummings in 1963 [1], [2] andindependently by Paul [3], as explicit solutions to the so-calledJaynes-Cummings-Paul Model (JCPM), for two-level emittersinteracting resonantly or non-resonantly with one radiationmode. Soon after, JCPM proved its usefulness in descriptionof spontaneous emission [4], and collapse and revival ofwavefunctions [5], [6]. Further development in experimentaltechniques and preparation of Rydberg atoms having verylarge transition dipole moments allowed various interactionregimes to be studied with regard to the magnitude of couplingconstant or Rabi frequency [7], [8]. According to the strengthof this constant, a CQED system may fall in three regimes:weakly, strongly, or ultrastrongly coupled. If the Rabi fre-quency is less than the decay rate of excited states in the cavitythen the interaction is usually weak, which happens to addressmost of the occurring CQED systems. Weakly coupled systemsare responsible for a number of well-known phenomena suchas enhanced or suppressed spontaneous radiation, which havefound applications in modern light emitting devices such assemiconductor lasers [9], [10].

Larger coupling constants exceeding one-fourth of the dif-ference of atom and field decay rates [11], enables the strongcoupling regime. Under strong coupling, the eignstates wouldbe no longer degenerate and recombine to form two separatestate pairs [12], [13], having an energy difference given bythe Rabi frequency. If the emitting system is a quantumdot or well, then these new quantum mixed states obtainedfrom photon-exciton interaction are sometimes viewed asnew quasi-particles referred to as exiton-polaritons. Stronglycoupled systems are building blocks of solid-state quantuminformation processing [14], and novel quantum phenom-ena including anti-bunching in single-photon emitters [15],quantum encryption [16], quantum repeaters [17], [18], andquantum computation [19].

Strong coupling in a cavity may be reached by increasingconfinement times and thus quality factors, while decreas-ing the effective mode volume [20]. At optical frequencies,semiconductor cavities [21], micro-disks [22] and photoniccrystal nano defects [23]–[27] have demonstrated successfuloperation of ultra-low threshold and polariton lasers, throughcombining excellent confinement and tight mode volumes.Another aspect of strongly-coupled CQED systems is controlof detuning frequency. High-fidelity single photon sources[28]–[31], high bandwidth low-threshold lasers [32]–[34],

0018-9197 © 2013 IEEE

ALIDOOSTY et al.: SIMULATION OF MULTIPARTITE CQEDs 1067

and single quantum-dot devices such as mirrors [35] and phaseshifters [36] are highly dependent on the possibility of controlon detuning. Among these, various methods such as cryogeniclattice temperature control [21], [37], [38], condensation atultralow temperatures [39], [40] and electrical control [41]may be mentioned.

The next interaction regime is the ultrastrong coupling,where the Rabi frequency is typically comparable or evenlarger than the decay rate [42]. This typically results inimproved excited and ground state properties such as non-adiabatic CQED phenomena [42]. Most ultrastrongly coupledsystems depend on the range of the radiation spectrum occurin either of the two solid state systems. At the optical fre-quencies, intersubband transitions of semiconductor cavitiesplaced in doped potential wells [43]–[47] form the physi-cally ultrastrong interaction. This is while at the microwavefrequencies, superconductor resonator circuits supercooled tomilli-Kelvin temperatures in resonance with two-level emittersformed from Josephson junctions may interact ultrastrongly[48], [49]. More recently at the Terahertz frequencies, a thirdultrastrongly coupled system has been identified [50], which isobtained from the interaction of magnetic cyclotron resonancesof a high-mobility two-dimensional electron gas in an ampli-fying medium. Since the cyclotron frequency, as the transitionfrequency, is very well controllable with a perpendicularmagnetic field, the transition dipoles are easily controlled andmay be increased to extremely high values. Rabi frequenciesas large as 58% of the transition frequencies have been sofar measured [50]. Also, metal-dielectric-metal microcavitiesalong with quantum wells have been shown to form idealsystems for generation of cavity polaritons at the Terahertzspectrum, in which Rabi frequencies exceeding 50% of thetransition frequency have been demonstrated [51]. Notably,it has been observed that under the ultrastrong coupling,significant deviations from classical behavior not foreseenin the standard context of quantum optics may arise. Theseinclude spontaneous conversion of virtual to real photons [51a]and breakdown of the well-known Purcell effect [51b].

All these three interaction regimes are supposed to bedescribable by a unified JCPM theory of quantum optics.Recently, we made an attempt to describe the most gen-eral CQED system [54] comprising an arbitrary number ofemitters and radiation modes subject to an arbitrary initialstate. Useful mathematical formulation and analysis of sucha system is highly contingent on a different and extendednotation of atomic and field states and operators, which wehad constructed therein. We furthermore have allowed field-dipole and dipole-dipole interactions to exist. Then the modelHamiltonian was transformed to the Heisenberg’s interactionpicture under Rotating Wave Approximation (RWA), and thesubsequent Rabi equations were numerically solved, as isroutinely done elsewhere, too [1].

However, as we are going to discuss it just below, thisgeneral approach is mathematically incorrect, especially forthe ultrastrongly coupled systems. And, this is not becauseof the RWA, but rather the Heisenberg’s transformationinvolved, which is employed in an incorrect manner. We aregoing to describe a universal approach, instead, which is

mathematically consistent and easy to deal with, and mayor may not include RWA. Moreover, our proposed methodleads to explicitly closed-form solutions, which may be rapidlyevaluated.

In the JCPM, the Hamiltonian is normally transformedinto the Heisenberg’s interaction picture [1], while it is takenas granted that the Bosonic field creation and annihilationoperators should obey simple first-order differential equationswith solutions varying in time as exp (±iωt). These free-running solutions for field operators clearly oscillate com-pletely sinusoidal in time. In our recent studies [55]–[58] ithas been noticed that these sinusoidal free-running solutionsfor field operators are not correct, and they in fact oscillatelargely non-sinusoidal. Consequently, not only RWA shouldbe avoided for such ultrastrongly coupled systems, but alsoHeisenberg’s transformation must be abandoned.

It is the purpose of this paper to construct and accuratelysolve a universal, self-consistent, and most general theoreticalpicture of multipartite CQED systems without RWA and/orHeisenberg’s transformation, while being applicable to ultra-strongly coupled systems. The explicit solution is greatlysimplified and accurately evaluated using algebraic matrixexponential techniques, as just reported in a recent publica-tion [58]. This technique allows rapid and accurate evalua-tion of state kets in time without relying on any numericalintegration. For treatment of partitions (or more accurately,particles) having spin, and in particular Fermions with half-integer spin, we may note that the corresponding spinorsare actually special cases of two-partite entangled systemsobtained by outer product of two scalar Fermions. Hence,the present formulation is equally capable of dealing withspin at the expense of a two-fold increase in the number ofscalar partitions. As a result, we have been able for the firsttime to investigate ultrastrongly coupled multipartite systemsnot been studied so far [59]. These include an ultrastronglycoupled integrated waveguide structure realized in compoundsemiconductor quantum wells [60], which is modeled as atwo-partition system consisting of a three-state �-emitter andone radiation modes. The other example is a seven-partitionsystem consisting of six identical two-level quantum dots andone radiation mode, having a maximum photon occupancynumber of twenty-four.

By plotting various probabilities and expectation values offield and atomic operators, we can find that ultrastronglycoupled systems are marked by onset of a chaotic behavior inphase space. Earlier last year [58] we had reported anomalousand largely nonlinear variations for phases of field operators.Our present study for the first time sets up a rigorous andnovel method to trace the evolution of multi-partite systemsin phase space, in which such chaotic behavior are easilydetectable. Our computer software code is theoretically capa-ble of dealing with any multipartite CQED system, and isable to self-generate an internal subroutine for exact calcu-lation of concurrence. This parameter represents the overalldegree of entanglement in a multipartite system. Finally, wepresent a complete numerical simulation of spin entanglementbetween a photon polarization and electron spin confined in asemiconductor quantum dot. We also demonstrate theoretically


how it would be possible to treat spin as an extra degree offreedom in a multi-partite quantum system.

II. MATHEMATICS AND ALGORITHM

To analyze the general behavior in CQED of complex multi-partite systems, initially the coefficients matrix of the mostgeneral system are computed and numerically measured. Suchsystems normally consist of an arbitrary number of emitters(usually quantum dots) in interaction with an arbitrary numberof cavity modes. For this purpose, the general time-dependentstate of the most general possible system has been rigorouslyspecified and is solved exactly in time-domain using an explicitanalytical solution presented in this section. As it will beshown, providing initial conditions as one of the parts ofthe solution is vital. Fock and coherent initial conditions areconsidered in this article, so the most general equation toprovide such coherent initial condition is extracted. Requiredequations to measure the presence probability of the systemat different states are presented. Expectation values of fieldand atomic operators as well as the expectation value ofcommutator of atomic ladder operators are also extracted.Finally an extensive high-level MATLAB code is developedwhich sets up the initial conditions for any arbitrary complexsystem and evaluates mentioned parameters including proba-bilities, expectation values of field and atomic operators, thecommutator of atomic ladder operators, as well as concurrenceas the most general measure of entanglement of multipartitesystems.

A. Coefficients Matrix

The aim of the developed code is to solve the Schrödingerequation given by [61]

∂

∂ t|ϕ (t)〉 = − i

hH |ϕ (t)〉 (1)

where |ϕ (t)〉 is the general state of system, H is the gen-eralized JCPM Hamiltonian presented in [54] and h is thereduced Plank constant. In Schrödinger space the ket statesof the system are time-dependent while operators are not.Due to the reasons discussed earlier [55]–[58], Heisenberg’stransformation must not be used.

We suppose that |A〉 is the ket state of the different energylevels of emitters, k is the total number of emitters, |F〉 is theket state of the cavity modes, fν is the number of photonsin ν–th cavity mode number, and ω is the total number ofcavity modes. The ket denoting the rn-th state of light emitting

system

∣∣∣∣

nrn

⟩

, expresses the condition that the n-th quantum

dot resides at its rn-th energy level. Now, the general time-dependent state of the most general possible system will begiven by

|ϕ(t)〉 =∑

A,F

φ (A, F) |A〉 |F〉

|A〉 =k⊗

n = 1

∣∣∣∣

nrn

⟩

=∣∣∣∣

1r1

⟩ ∣∣∣∣

2r2

⟩

· · ·∣∣∣∣

krk

⟩

1 ≤ rn ≤ Bn

|F〉 =ω⊗

υ = 1| fν 〉 = | f1〉 | f2〉 . . . | fω〉 0 ≤ fν ≤ Nν (2)

where rn refers to the different energy level states of lightemitting systems and Bn is the maximum number of energylevels of the n-th quantum dot. Nν is the number of maximumphotons which possibly occupies a cavity mode [54], andhere is taken to be identical for all modes. Also, |ϕ(t)〉 isa superposition of all possible states of the system, includingatom and field states, and each state has a time dependentcoefficient equal to φ (A, F).

So according to (2), various states are formed by thedifferent states of quantum dots at different energy levelsmultiplied by different photon number states in each cavitymode. If m is the total number of cavity modes, then (2) canbe written as

|ϕ(t)〉 =Bn∑

r1,r2, ...,rk=1

Nν∑

f1, f2, ... , fν , ..., fω=0

{φ (r1, r2, . . . , rk , f1, f2, . . . , fν, . . . , fω) |A〉 |F〉} (3)

The generalized JCPM Hamiltonian H is here consisting ofthree parts as [54], [58]

H = H0 + Hr·E + Hr·r (4)

H0 =∑

n,i

Eni σ

ni +

∑

ν

hν a†ν aν (5)

Hr·E =∑

n,i< j

(

γni j σni, j + γ ∗

ni j σnj,i

)

×∑

ν

(

gni j aν + γ ∗ni jν a†

ν

)

(6)

Hr·r =∑

n<m,i< j

{(

ηni j σni, j + η∗

ni j σnj,i

) (

ηmi j σmi, j + η∗

mi j σmj,i

)}

(7)

in which H0 describes the system energy without interaction,Hr·E stands for light-emitter interactions, and Hr·r representsinteractions between any possible pair of emitters such asdipole-dipole terms [54], [58]. Coefficients γni j are matrixelements of dipole operator of n-th emitter. The strength ofthe dipole interaction between n-th emitter and ν-th mode ofcavity is given by gni jν with the transition i -th and j -th energylevels. Coefficients ηni j are proportional to the strength of thedipole generated while another emitter undergoes a transitionbetween i -th and j -th levels. En

i indicates the i -th energy ofthe n-th emitter. Furthermore a†

ν and aν are the field creationand annihilation operators which respectively increase anddecrease the number of existing photons within the ν-th cavitymode by one, σ l

s,k is atomic ladder operator which makes the

l-th emitter to switch from k-th to s-th level, and(

σ ls,k

)† =σ l

k,s is its adjoint.By equalizing the coefficients of similar kets on both sides

of (1), the coefficients may be rearranged as elements of asquare matrix with dimension N as [M]N×N [18] as

∂

∂ t{ϕ (t)}N×1 = [M]N×N {ϕ (t)}N×1 (8)

in which {ϕ(t)} is the vector of unknown coefficients. Wesuppose that {ϕ(0)} is the initial condition vector. Then thesolution to (8) is simply given by

{ϕ (t)} = e[M] t {ϕ (0)} (9)


To evaluate (9), [M] is first diagonalized into a diagonal matrix[D] = [Diδi j ] of eigenvalues Di using the diagonalizer [R]which is found from eigenvectors of [M] as

{ϕ (t)} = e[M]t {ϕ (0)} = [R]e[D]t [R]−1 {ϕ (0)}= [R]

[

eDi tδi, j

]

[R]−1 {ϕ (0)} (10)

The solution (10) is explicit and can be accurately evaluatedregardless of the time t . It also excludes the need of matrixexponentiation and is thus numerically stable. We also notethat {ϕ (0)} can take on any initial such as Fock or coherent[62] initial states. For the most general multipartite systemconsisting of an arbitrary number of modes and emitters, thenormalized initial coherent states will be

|ϕ(0)〉=⎧

⎨

⎩

ω∑

l=1

e−|λl |2Nl∑

αl=0

|λl |2αl

αl !

⎫

⎬

⎭

−1/2

×ω∏

j=1

e− 1

2|λ j |2

N j∑

n j =0

λn jj

√

n j !|n1, . . . , nω〉 |energy states〉

(11)

It must be here added that truncation to the dimension Nis in general valid (comparison to a real experiment laterin Section IV would justify this picture), but only wheneverthe dipole and field interactions vanish at high energies.Otherwise, one would need to take care of all numericallysignificant interactions regardless of the initial conditions, andof course without such a truncation.

B. Probabilities of Presence at States

According to (3), the presence probability of an arbitrarylight emitting system such as l, being in at arbitrary energylevel such as m is simply

P =Bn∑

A−{rl }=1

×⎧

⎨

⎩

Nν∑

f1, f2,..., fω=0

|ϕ (r1, r2, . . . , rl→m , rk , f1, . . . fω)|2⎫

⎬

⎭

(12)

C. Expectation Values of Field Operators

Expectation values of annihilation operator for the mostgeneral possible complex system are found as

〈ϕ(t)| aν |ϕ(t)〉 =∑

A,F

√

fνφ∗ (A, fν − 1) φ (A, fν)

=Bn∑

r1,r2,...,rk=1

Nν∑

f1, f2,..., fν ,..., fω=0

{√

fν

φ∗ (r1, r2, . . . , rk , f1, f2, . . . , fν − 1 . . . , fω)

×φ (r1, r2, . . . , rk, f1, f2, . . . , fν, . . . , fω)}

(13)

D. Expectation Values of Ladder Operators

Expectation values of the atomic transition operator arefound as

〈ϕ(t)| σ ls,m |ϕ(t)〉

=∑

A−{rl },F

{(

φ∗ (

Arl→s , F) ⟨

Arl→s∣∣ 〈F |)

·(φ (

Arl→m, F) ∣∣Arl→m

⟩ |F〉)}=

∑

A−{rl },Fφ∗ (

Arl→s , F)

φ(

Arl→m F)

=Bn∑

A−{rl }=1

{ Nν∑

f1, f2,..., fω=0

φ∗ (r1, r2, rl → s,. . ., rk , f1, f2,. . ., fω)

×φ (r1, r2, rl → m,. . ., rk, f1, f2,. . ., fν ,. . ., fω)

}

(14)

E. Expectation Value of the Ladder Commutator

The commutator of the atomic ladder operators is thecommutation of atomic transition operator and its Hermitianadjoint, denoted by [σ l

s,m, σls,m

†], with the expectation valuefound as

〈ϕ(t)|[

σ ls,m,

(

σ ls,m

)†]

|ϕ(t)〉=

∑

A−{rl },F

∣∣φ∗(Arl→s , F

)∣∣2 −

∑

A−{rl },F

∣∣φ∗ (

Arl→m , F)∣∣2. (15)

III. ANALYSIS AND NUMERICAL RESULTS

Here, we present detailed analysis of two different quantumoptical systems: a three-level quantum well and a multipartitequantum optical system with seven partitions. All extractedequations in the previous section by the utilization of ourprovided code are executed. Entanglement is also analyzedin both systems.

A. CQED in an Optoelectronic Device

In this section, we present the simulation and analysis ofthe CQED of a real complex system consisting of a threelevel light emitting system interacting with a cavity mode.The emitter is an InGaAlAs quantum well and the light isguided in a waveguide underneath [59]. The design detailsand applications of such optoelectronic device as a wide-bandand ultra-compact optical modulator is discussed elsewhere[59]. It has been shown that for the system of interest, thequantum well could be modeled as a three-level light emittingsystem with defined energy levels, corresponding to electrons,and heavy and light holes bands as in Fig. 1. Transition dipolemoments between different energy levels are also calculatedin [59].

1) System Characteristics: The light emitting systemdescribed in our other article [59] is a heterostructure In0.52(AlxGa1−x)As/In0.53Ga0.47As/In0.52(AlxGa1−x)0.48As quan-tum potential well. For an Aluminum fraction of x = 0.9 andwell material thickness of 9 nm, the transition energy betweenconduction and heavy hole bands will be about 0.8 eV. Thereis a 30 meV offset between the heavy and light holes bands.We thus may choose the energy levels of the model light


Fig. 1. QW with the electron, heavy-hole and light-hole levels [57].

TABLE I

RABI FREQUENCIES AND COUPLING REGIMES

emitter system respectively as 0, 30, and 829 meV for lightholes, heavy holes, and electrons bands. Rabi frequencies arecalculated as

Gh = 1

h(E0 E) 〈ψe |eR |ψhh〉 (16)

Gl = 1

h(E0 E) 〈ψe |eR |ψlh〉 (17)

in which Gh and Gl are the Rabi frequencies for transitionbetween conduction band to heavy holes and light holes,respectively. 〈ψe |eR |ψhh〉 and 〈ψe |eR |ψlh〉 are transitiondipole moments, respectively calculated as 26.15 Debye and15.21 Debye [59], and E0 E is applied electrical field. SinceRabi frequencies (16) and (17) are functions of the appliedfields, field-emitter interactions are fundamentally nonlinear.Evidently, these nonlinearities are neglected in our presentformulation, which may eventually result in excitation ofunwanted optical modes (both confined or evanescent) andsubsequent undesired energy coupling, decoherence, and loss.

Table I shows the range of Rabi frequencies measured basedon (16) and (17) for various electric field strengths. It wasassumed that the light emitting system is in interaction withphotons having the wavelength of λ = 1.49μm so the opticalfrequency ωλ would be equal to 1.2582 × 1015 rad/s.

To simulate different coupling regimes, electrical field isallowed to vary while the features of the considered system arefixed. According to the Table I, and because of near-resonancesituation between light and electron-heavy hole transition, inthe weakest applied electric field with E0 = 100 kV/cm, thesystem is in the weak coupling regime. For a stronger electricfield with E0 = 1MV/cm, the coupling regime is strong.

Finally, for the strongest electric field with E0 = 10MV/cm,the coupling enters the ultrastrong regime.

The general time-dependent state of the system and itsdescribing Hamiltonian according to the Eqs. (2), (5)–(7) arenow expressed as

|ϕ(t)〉 =∑

A=lh,hh,e,F

φ (A, F) |A〉| F〉 (18)

H0 =∑

1,i

E1i σ

1i + ha†a

Hr·E =∑

n,i< j

(

γni j σ1i, j +γ ∗

ni j σ1j,i

)∑

ν

(

gni j a+g∗ni j a

†)

(19)

In this system, Hr·r is equal to zero because there is only oneemitter. As input of our software all coupling coefficients areentered in units of energy. The coherent initial state in thissystem follows (11).

2) Presence Probabilities With Fock Initial State: By apply-ing Fock initial conditions we intended to study two cases.Firstly, to study the effect of boosting coupling coefficienton the presence probability of the system in normalizedtime; secondly, to study the importance of coefficient matrixmeasurement of the system exactly and without RWA. As weknow, in RWA the effect of two terms σ 1

i, j†a† = σ 1

j,i a† and

σ 1i, j aν are neglected.Since the light emitting system under consideration consists

of three energy levels, the effect of both σe,lh a† and σe,hh a†

could be studied. To study the σe,lh a† term due to Fock initialcondition, |1, lh〉 state is considered. For instance under theoperation of σe,lh a† the state ket will become√

2 γ1,lh,hh g∗1,lh,hh |2, hh〉 + √

2 γ ∗1,lh,e g∗

1,lh,e |2, e〉 (20)

Since, heavy to light holes transition is forbidden, we haveg1,lh,hh = 0, and hence only the probability presence of thesystem in the |2, e〉 state should be calculated, which is plottedfor |2, e〉 and |1, lh〉 as a function of normalized time differentcoupling regimes in Figs. 2 and 3. The effect of σe,hha† termmay be studied similarly.

As it is here seen, by increasing coupling strength, thefrequency of oscillations also increases, except for the ultra-strong regime which exhibits a non-sinusoidal and disorderedbehavior or anonymous oscillations. It is also observed thatalthough the probability of the states which are neglected inRWA, is negligibly small in weak and strong coupling, while itis significantly larger in the ultrastrong coupling. Hence, RWAis a very inappropriate approximation for study of ultrastrongcoupling [64].

3) Presence Probabilities With Coherent Initial State: Fol-lowing (12), the presence probability of system at each of theconduction, heavy hole, or light hole levels. This probabilityis also found and plotted in the Fig. 4. By comparison itis observed that in the weak and strong coupling regimes,the presence probability of the system in different states issinusoidal on short time scales. The probability for a maximumphoton occupancy number of 8 is

P =8

∑

f1=0

|ϕ (r1→e, f1)|2 (21)


Fig. 2. The presence probability of the system in |1, lh〉 under (a) weak,(b) strong and (c) ultrastrong coupling.

Fig. 3. The presence probability of the system in |2, e〉 under (a) weak,(b) strong and (c) ultrastrong coupling regimes.

This is while by entering into ultracoupling regimes, thebehavior is not sinusoidal and has anonymous oscillations atall on any time scale and is very chaotic both in short and longtemporal range. Remarkably, nonlinear quantum optical chaosin an optical beam has been observed recently in experiment[63] through direct measurement of the evolution of Wignerfunction in phase space.

All calculations are here done within the standard double-precision numerical accuracy, and hence so are the eigen-values, which in this example come from small matrixdimensions. Therefore, we may conclude that the chaoticdynamics is not a result of systematic or round-off numer-ical errors. Indeed a recent separate study of ours verifiesthe fact that the so-called Kolmogorov entropy indeed isa monotonic increasing function versus coupling strength,

Fig. 4. Probabilities of occupation of the, light-hole, lh, heavy-hole, hhand conduction, e states (blue, red and black in each figure respectively);(a) weakly, (b) strongly, and (c) ultrastrongly coupled systems.

thus numerically proving the emergence of the chaoticdynamics [65].

We furthermore ignore the cavity decay by assuming alarge quality factor, which physically corresponds to a longconfinement time. Quality factors in excess of 106 havebecome feasible in nanophotonic cavities [66], which typicallycorrespond to 106 cycles of light oscillations. For all practicalreasons, such extraordinary confinements defy leakage ofoptical energy within the simulation time. We also ignoredecoherence mechanisms, which in principle should be addedto the Hamiltonian as separate terms using the incorporationof superoperators [67]. In general, simulation of decoherencein multipartite systems is considered to be too complex andnumerically demanding, which hopefully will be addressed infuture.

4) Annihilation in Different Coupling Regimes: Following(13), the expectation value of the field annihilation operatorof the system having coherent initial state is measured. Inorder to study the behavior of field operators in different cou-pling regimes, this expression has been calculated. Since theannihilator is non-Hermitian, its expectation value is complex-valued. Making a three-dimensional parametric plot havingthe corresponding real and imaginary parts as functions ofnormalized time is very instructive in this case. This has beenshown for various coupling regimes and shown in Figs. 5and 6. Also the phase of expectation value as a function ofnormalized time duration has been also plotted in separatediagrams therein.

It is observed that in the weak coupling regime the behaviorof the expectation value of the field annihilation operator iscompletely sinusoidal and it confirms that solving Schrödingerequation in Heisenberg space with RWA can be acceptablein this regime. By increasing the coupling constant andentering into strong regime, as it is seen in Fig. 5, thebehavior keeps varying sinusoidally, but the amplitude ofoscillations gradually decrease with time. This is while the


Fig. 5. Three-dimensional plots of the real and imaginary values of theexpectation value of a in (a) weakly, (b) strongly and (c) ultrastrongly coupledsystem.

Fig. 6. Phase plots of the real and imaginary values of the expectation valueof a in (a) weakly, (b) strongly and (c) ultrastrongly coupled systems.

phase behaves just similarly to the weak coupling regime.For the ultrastrongly coupled system, as shown in Fig. 5, how-ever, the expectation value plot is remarkably non-sinusoidaland chaotic and has anonymous oscillations. At the same time,the phase is significantly non-linear. We had also noticed thisparticular behavior of the phase under ultrastrong coupling inour recent studies [55]–[58].

5) Expectation Value of the Ladder Operator: Accordingto (14), the expectation value of the atomic ladder operatorfor transitions between conduction and heavy and light holeswhile initial state is coherent are measured. These have beencalculated and plotted in Figs. 7–10, as parametric plots similarto those of annihilator operator in the above. Similarly, asthe system enters the ultrastrong coupling regime, chaoticbehavior starts to develop, which is clearly visible both inthe phase space and phase plot.

Fig. 7. Three-dimensional plots of the real and imaginary values of theexpectation value of σhh,e in (a) weakly, (b) strongly and (c) ultrastronglycoupled system.

Fig. 8. Phase plots of the real and imaginary values of the expectation valueof σe,hh in (a) weakly, (b) strongly and (c) ultrastrongly coupled systems.

6) Entanglement: The expectation value of the commutatorof atomic ladder operators is known for simple quantumsystems to encompass basic information regarding the degreeof entanglement. For multipartite systems, however, a morecomplicated measure such as concurrence [68] should becomputed. For the system under study, we calculate and plotboth. For an initial coherent state, the expectation of com-mutator of the ladder operators (15) has been calculated andplotted for various transitions in Figs. 11 and 12, respectively,for hh-e, and lh-e transitions. Again the general trend issuch that the strongly coupled system exhibits fast and slowcomponents multiplied together, while this differentiation offast and slow oscillations in the ultrastrongly coupled systemsis not possible.

By recycling the computer software provided in our previ-ous research [58], which self-generates an internal subroutinefor exact calculation of concurrence, we were also able to com-pute graphs of concurrence under various coupling strengths,ranging from weak to strong and ultrastrong regimes. This hasbeen shown in Fig. 13.


Fig. 9. Three-dimensional plots of the real and imaginary values of theexpectation value of σlh,e in (a) weakly, (b) strongly and (c) ultrastronglycoupled systems.

Fig. 10. Phase plots of the real and imaginary values of the expectation valueof σlh,e in (a) weakly, (b) strongly and (c) ultrastrongly coupled systems.

It is here concluded through observations made on manymultipartite quantum systems that the entanglement of inall cases behaves also chaotic in the ultrastrong regime andexhibits lots of disordered oscillations and distortion. Furtherdetails of calculations and plots as well as program sourcecodes are presented for this system and other examples else-where [60].

B. CQED in a Seven-Partition System

In this section we report the simulation and analysis of aseven-partition system consisting of identical six quantum dotsinteracting with one cavity mode. Due to the larger number ofsystem partitions, the number of different states of the systemincreases very much.

1) System Specifications: The six quantum dots are herelimited to two ground and excited states each, with energy

Fig. 11. Expectation value of[

σhh,e,(

σhh,e)†

]

in (a) weakly, (b) strongly,and (c) ultrastrongly coupled systems.


σlh,e(σlh,e)†]

in (a) weakly, (b) strongly,and (c) ultrastrongly coupled systems.

eigenvalues of 0 and 1 eV, respectively. The condition onidentically of dots may not be achieved in practice, and thedeveloped software code is able to treat different emitters withequal efficiency. The condition is set here for simplificationof the problem and reduction of the too many degrees offreedom. It is furthermore supposed that transition dipolemoment in these quantum dots is 192 Debye. We also allowmutual dipole-dipole interactions between all the dots witha magnitude of 5meV. It is also assumed that quantum dotsare interacting with one resonant cavity mode, having thefrequency

ωλ = Ei − Eg

h= 1 eV

h= 1.5177 × 1015rad/s (22)

By applying different electrical fields and comparing withthe optical frequency ωλ, different coupling regimes aresimulated. According to (16) and (17), Rabi frequencies


Fig. 13. Computed graphs of concurrency, (a) weakly, (b) strongly, and(c) ultrastrongly coupled systems.

TABLE II

RABI FREQUENCIES AND COUPLING REGIMES

as the coupling constants are here calculated and enlistedin Table II, for weakly, strongly, and ultrastrongly coupledsystems.

With the assumption that the maximum possible number ofphotons in the cavity mode is 8, the general time dependentstate of system and the describing Hamiltonian by (2), (5), (6),and (7) were specified. Initial Fock and coherent conditions(11) were considered to simulate the system.

2) Presence Probabilities With Fock Initial State: We firstassume that the initial state is simply at a Fock-state, whichexpresses that there is exactly one photon in the cavity modeand all quantum dots are in their ground energy level. Settingthis ket as the initial state, we calculate and plot the presenceprobability in this state as a function of normalized time indifferent coupling regimes in Fig. 14.

As it is seen in these plots, by increasing the couplingconstant the oscillation frequency increases, while in theultrastrong regime the behavior is chaotic and undergoesanonymous oscillations. This characteristic behavior of theultrastrong coupling is also justified similarly in the rest ofsimulations, as discussed in the following.

3) Presence Probability With Coherent Initial State: Thepresence probability of the first dot being in its ground,

Fig. 14. The presence probability of the system in |1, g, g, g, g, g, g〉 ketstate under weak, strong and ultrastrong coupling regimes from left to rightand below respectively.

Fig. 15. Probabilities of occupation of the excited and ground energylevel states in the sixth quantum dot (red and blue colors in each figurerespectively); from left to right: weakly, strongly and ultrastrongly coupledsystem.

or excited energy levels is given according to (12) by

Pg =∑

A−{r1 }=g

8∑

f1=0

∣∣ϕ

(

r1→g, r2, . . . , r8, f1)∣∣2

Pe =∑

A−{r1}=e

8∑

f1=0

|ϕ (r1→e, r2, . . . , r8, f1)|2 (23)

Similar expressions may be obviously written for each of thequantum dots. These probabilities for the sixth dot have beenplotted in Fig. 15, for weak, strong, and ultrastrong coupling.The characteristic chaotic behavior of ultrastrong coupling canbe seen again.

4) Annihilation in Different Coupling Regimes: The expec-tation value of the field annihilation operator of the system ismeasured based on (13), due to initial coherent state. Phasespace and phase plots of the real and imaginary values of the


Fig. 16. Three-dimensional plots of the real and imaginary values of theexpectation value of a in (a) weakly, (b) strongly and (b) ultrastrongly coupledsystem.

Fig. 17. Phase plots of the real and imaginary values of the expectationvalue of a in (a) weakly, (b) strongly and (c) ultrastrongly coupled systems.

expectation value as functions of normalized time are shownin Figs. 16 and 17, respectively for weakly, strongly, andultrastrongly coupled systems.

It is surprisingly observed that by the increasing the cou-pling constant the entering into ultrastrong regime the cou-pled system not only exhibits a very chaotic and disorderedbehavior in the three-dimensional parametric plot, but alsothe corresponding phase changes abruptly. This behavior isalso seen in nearly all other complex expectation values of allultrastrongly coupled multipartite systems we have studied sofar, and is yet to be understood.

5) Expectation Value of the Atomic Ladder Operator: Againwe choose the sixth dot as the illustrative example. Accordingto (14), the expectation value of the atomic ladder operatorfor decay of every quantum dot individually is measuredwhile initial state is coherent. Three-dimensional (phase space)and phase plot of the real and imaginary values of this

Fig. 18. Three-dimensional plots of the real and imaginary values theexpectation value of σ for the sixth quantum dot in weakly, strongly andultrastrongly coupled system from left to right.

Fig. 19. Phase plots of the real and imaginary values of the expectation valueof σ for the sixth quantum dot in (a) weakly, (b) strongly and (c) ultrastronglycoupled system.

expectation value as functions of normalized time duration hasbeen similarly plotted in Figs. 18 and 19. As it is observedin numerical simulations, all six dots behave more or lessaccording to the same pattern with slight differences are seenin the whole system.

All six quantum dots have the same sinusoidal or nearly-sinusoidal oscillations in weakly and strongly coupled systems,respectively. This is while the ultrastrong coupling is accompa-nied with chaotic three-dimensional trajectories and multi-steprandom-like abrupt phase changes for all six quantum dots.These abrupt phase changes may find applications in multi-state quantum information processing later, if understood andpredicted correctly.

6) Entanglement: The expectation value of the commutatorof atomic ladder operators for transition of every quantum dotindividually from excited energy level to ground energy levelhas been also analyzed as given by (15). Plots are presented inFig. 20. It is observed and concluded by the measurements thatthe system entanglement in all quantum dots is chaotic underthe ultrastrong regime and accompanied by a lot of distortion.



σg,e(σg,e)†]

in (a) weakly, (b) strongly, and(c) ultrastrongly coupled systems.

IV. SPIN ENTANGLEMENT

In this section, we offer a method to deal with the sys-tems arising from spin interactions. Such processes includingspin-photon entanglement have just recently been observed[69], [70]. In order to treat the effect of spin, we rewrite thesystem state and base kets as

|ϕ(t)〉 =∑

A,F

φ (A, F) |A ↑〉 |A ↓〉 |F ↑〉 |F ↓〉

|A ↑〉 =k⊗

n =1

∣∣∣∣

nrn

↑⟩

=∣∣∣∣

1r1

↑⟩ ∣∣∣∣

2r2

↑⟩

· · ·∣∣∣∣

krk

↑⟩

1 ≤ rn ≤ Bn

|A ↓〉 =k⊗

n =1

∣∣∣∣

nrn

↓⟩

=∣∣∣∣

1r1

↓⟩ ∣∣∣∣

2r2

↓⟩

· · ·∣∣∣∣

krk

↓⟩

|F ↑〉 =ω⊗ν=1

| fν ↑〉=| f1 ↑〉 | f2 ↑〉 · · · | fω ↑〉 0 ≤ fν ≤ Nν

|F ↓〉 =ω⊗ν=1

| fν ↓〉=| f1 ↓〉 | f2 ↓〉 · · · | fω ↓〉 (24)

In (24) |A ↑〉 and |A ↓〉 denotes the ket states of the differentenergy levels of the two systems being formed from the ketstate, |A〉 due to the presence of an external magnetic field.So according to (24), we allow all emitters indexed by nto take on up or down spin states. Similarly, cavity photonFock states indexed by ν can take on up or down states(corresponding actually to the two counter-rotating circularpolarizations of photons), |F ↑〉 and |F ↓〉. Other parametersare those introduced in (2). Expansion coefficients will betherefore given by

|ϕ(t) 〉 =Bn∑

r1,r2,...,rk=1

Nν∑

f1, f2,..., fν ,..., fω=0{

φ(

r1↑, r1↓, · · · , rk↑, rk↓, f1↑, f1↓, . . . , fω↑, fω↓)}

(25)

Fig. 21. (a) Presence probability and (b) concurrency versus time arecalculated for (c) two coupled �-emitter systems; (d) superimposed numericalsimulation (blue) versus experimental data (green) from [69].

The generalized JCPM Hamiltonian H should be also similarlyupdated

H = H0 + Hr·E + Hr·r (26)

H0 =∑

n,i

(

Eni↑σ n

i↑ + Eni↓σ n

i↓)

+∑

ν

(

hν↑a†ν↑aν↑ + hν↓a†

ν↓aν↓)

(27)

Hr·E =∑

n,i< j

(

γni j σni↑, j↓ + γ ∗

ni j σnj↓,i↑

)

×∑

ν

(

gni jν↑aν↑ + g∗ni jν↑a†

ν↑)

+∑

n,i< j

(

γni j σni↓, j↑ + γ ∗

ni j σnj↑,i↓

)

×∑

ν

(

gni jν↓aν↓ + g∗ni jν↓a†

ν↓)

(28)

Hr·r =∑

n<m

{(

ηni j σni↑, j↑ + αni j σ

ni↓, j↓ + δni j σ

ni↑, j↓

)

×(

ηmi j σmi↑, j↑ + αmi j σ

mi↓, j↓ + δmi j σ

mi↑, j↓

)}

(29)

In the first interaction term Hr·E we note that any atomictransition due to interaction with photon should accompany bya unit change in photon spin. Hence those terms violating theconservation of total spin have been dropped. In the absenceof an external magnetic field (or any symmetry breakingperturbation), states having up or down states are doubledegenerate and the system may be effectively simplified to thatof the interaction picture described in Section II. Otherwise,the above sets of equations are minimally needed to fullyaccount for the effect of spin interaction terms.

Now, we proceed in the following to present a full numericalsimulation of electron-photon spin entanglement effect.


A. Full Numerical Simulation of Spin Entanglement

As an example of our methodology, the matter photonentanglement observed recently in [69] is simulated here.Following the reported experiment in [69], a quantum opticalsystem consisting of a two-level emitter with two energystates, embedded in a planar photonic micro-cavity, is placedin interaction with a cavity mode occupied with one photon.The emitter is an InAs quantum-dot, that due to the presence ofan external magnetic field applied perpendicularly to its growthdirection is turned into two coupled L-emitter systems. Thesetwo emitters consist of separate excited and ground energystates as depicted in Fig. 21. Each of the excited states ortrion states have two paired electrons and an unpaired hole,|↑↓⇓〉 and |↑↓⇑〉, coupled to two ground states which havespin states |↑〉 and |↓〉 as shown in Fig. 21. The emitters areassumed to be in interaction with a photonic qubit (H and Vpolarized photons) at the wavelength of 910.10 nm. In [69] theenergy difference between two emitters due to a magnetic fieldof 3T corresponds to an angular frequency of 2π×17.6 GHz.To simulate this system, first by following (24) the ket stateof the system is specified as

|ϕ(t)〉 =∑

A↑=g↑,g↓,e↑↓⇑

∑

A↓=g↓,g↑,e↑↓⇓1

∑

F1↑=0

1∑

F1↓=0

φ (A, F) |A ↑〉 |A ↓〉 |F1 ↑〉 |F1 ↓〉 (30)

Following (25) the describing Hamiltonian for this system isgiven as

H0 =∑

1,i=g↑,g↓,e↑↓⇑,e↑↓⇓

(

E1i↑σ 1

i↑ + E1i↓σ 1

i↓)

+∑

ν

(

hν↑a†ν↑aν↑ + hν↓a†

ν↓aν↓)

(31)

Hr·E =∑

1,i=g↑,g↓,< j=e↑↓⇑,e↑↓⇓

{(

γ1i j σ1i↑, j↓ + γ ∗

1i j σ1j↓,i↑

)

×(

g1i jν↑aν↑ + g∗1i jν↑a†

ν↑)}

+∑

1,i< j

{(

γ1i j σ1i↓, j↑ + γ ∗

1i j σ1j↑,i↓

)

×(

g1i jν↓aν↓ + g∗1i jν↓a†

ν↓)}

(32)

in which i, j denote the energy states, in such a way thatin any term i refers to an energy level lower than that of j .ν ↑ and ν ↓ refer to the cavity modes occupied by H andV polarized photons, respectively. Finally, Hr·r has to beset to zero because it describes the interaction between twophysically separate emitters (e.g. in two quantum dots), andthere is no such interaction term.

Now, by using the Hamiltonian defined in (31) and (32),and following the method of solving Schrödinger’s equation(1) similar to what has been done in the previous section, thecoefficient matrix and time-dependent ket of the system canbe computed without any approximation. In this simulation,initially the system is supposed to be in the ground state,with the spin down and up interacting with one cavity modeoccupied by the H polarized photon in weak coupling regime.

The presence probability of the system being in this ket stateis measured and plotted in Fig. 21 according to (12). In orderto study the entanglement, the concurrency parameter is alsocomputed as explained extensively elsewhere [58] and [68].

As it was expected from experimental observations [69],[70] and using a fitting coupling strength of 7meV, we cansimulate a sinusoidal behavior for the probability with a periodof 50ps, in agreement to the experimentally observed value of50 ± 5ps [69]. Based on the simulation data, we believe thatthe interaction regime of the reported experiment falls in thestrong coupling. Similarly, the variations of concurrency alsoattain the same time-scale of oscillations. Nearly sinusoidaloscillations of this parameter confirm the periodic build-up anddestruction of entanglement between quantum-dot’s electronspin and photon’s polarization. Finally, the range of probablityvariations in Fig. 21(a) is found to be 0.25 to 1.0, whereasexperimental values are 0.20 ± 0.05 to 1.0 [69].

V. CONCLUSION

In this paper, the general behavior of CQED of com-plex systems under different coupling regimes was analyzed.Mathematically we tackled a general quantum optical systemconsisting of an arbitrary number of light emitters interactingwith an arbitrary number of cavity modes. We presented howto specify the time dependent state of the system, provideinitial conditions, and to solve the system in time-domain inSchrödinger picture, within the approximation of vanishinghigh energy field and dipole interactions. Next, we presentedexpressions for measuring presence probabilities, expectationvalue of field operators, atomic operators, and commutators.We have developed an extensive MATLAB code to producethe necessary initial conditions and solve the system. We alsopresented and discussed two example systems in details. Weconfirmed that RWA is invalid in ultrastrong coupling, andfurthermore observed, for the first time, a chaotic behaviorin ultrastrong coupling regime accompanied by multi-step andrandom-like abrupt phase changes. In the end, we successfullyreproduced the results of a recent spin-photon entanglementexperiment.

REFERENCES

[1] W. P. Schleich, Quantum Optics in Phase Space, 1st ed. Berlin,Germany: Wiley, 2001.

[2] E. T. Jaynes and F. W. Cummings, “Comparison of quantum andsemiclassical radiation theories with application to the beam maser,”Proc. IEEE, vol. 51, no. 1, pp. 89–109, Jan. 1963.

[3] H. Paul, “Induzierte emission bei starker einstrahlung,” Ann. Phys.,vol. 466, nos. 7–8, pp. 411–412, 1963.

[4] F. W. Cummings, “Stimulated emission of radiation in a single mode,”Phys. Rev., vol. 140, no. 4A, pp. A1051–A1056, 1965.

[5] J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodicspontaneous collapse and revival in a simple quantum model,” Phys.Rev. Lett., vol. 44, no. 20, pp. 1323–1326, 1980.

[6] G. Rempe, H. Walther, and N. Klein, “Observation of quantum collapseand revival in a one-atom maser,” Phys. Rev. Lett., vol. 58, pp. 353–356,1987.

[7] S. Haroche and J. M. Raimond, “Radiative properties of Rydberg statesin resonant cavities,” in Advances in Atomic and Molecular Physics,vol. 20, D. Bates and B. Bederson, Eds. San Francisco, CA, USA:Academic, 1985, pp. 347–411.


[8] J. A. C. Gallas, G. Leuchs, H. Walther, and H. Figger, “Rydbergatoms: High-resolution spectroscopy and radiation interaction—Rydbergmolecules,” in Advances in Atomic and Molecular Physics, vol. 20,D. Bates and B. Bederson, Eds. San Francisco, CA, USA: Academic,1985, pp. 413–466.

[9] V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov,and S. R. Forrest, “Weak microcavity effects in organic light-emittingdevices,” Phys. Rev. B, vol. 58, no. 7, pp. 3730–3740, 1998.

[10] R. B. Fletcher, D. G. Lidzey, D. D. C. Bradley, M. Bernius,and S. Walker, “Spectral properties of resonant-cavity, polyfluorenelight-emitting diodes,” Appl. Phys. Lett., vol. 77, pp. 1262–1264,Jun. 2000.

[11] D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler,M. Kamp et al., “Photon antibunching from a single quantum-dotmicrocavity system in the strong coupling regime,” Phys. Rev. Lett.,vol. 98, no. 11, pp. 117402–117405, 2007.

[12] E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard,et al., “Exciton-photon strong-coupling regime for a single quantumdot embedded in a microcavity,” Phys. Rev. Lett., vol. 95, no. 6,pp. 067401-1–067401-4, 2005.

[13] J. P. Reithmaier, “Strong exciton–photon coupling in semiconduc-tor quantum dot systems,” Semicond. Sci. Technol., vol. 23, no. 12,pp. 123001-1–123001-18, 2008.

[14] W. P. Schleich and H. Walther, Elements of Quantum Information.Weinheim, Germany: Wiley-VCH, 2007.

[15] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, andH. J. Kimble, “Photon blockade in an optical cavity with one trappedatom,” Nature, vol. 436, pp. 87–90, Jul. 2005.

[16] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptogra-phy,” Rev. Mod. Phys., vol. 74, pp. 145–195, Mar. 2002.

[17] E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficientquantum computation with linear optics,” Nature, vol. 409, pp. 46–52,Jan. 2001.

[18] Y. Yamamoto, F. Tassone, and H. Cao, Semiconductor Cavity QuantumElectrodynamics. Berlin, Germany: Springer-Verlag, 2000.

[19] W. Dür, H.-J. Briegel, J. I. Cirac, and P. Zoller, “Quantum repeatersbased on entanglement purification,” Phys. Rev. A, vol. 59, no. 1,pp. 169–181, 1999.

[20] S. E. Morin, C. C. Yu, and T. W. Mossberg, “Strong atom-cavity cou-pling over large volumes and the observation of subnatural intracavityatomic linewidths,” Phys. Rev. Lett., vol. 73, no. 11, pp. 1489–1492,1994.

[21] J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzen-stein, et al., “Strong coupling in a single quantum dot-semiconductormicrocavity system,” Nature, vol. 432, pp. 197–200, Nov. 2004.

[22] H. Cao, J. Y. Xu, W. H. Xiang, Y. Ma, S.-H. Chang, S. T. Ho, et al.,“Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys.Lett., vol. 76, pp. 3519–3521, Apr. 2000.

[23] R. Buttéa, G. Christmann, E. Feltin, A. Castiglia, J. Levrat, G. Cosendey,et al., “Room temperature polariton lasing in III-nitride microcav-ities: A comparison with blue GaN-based vertical cavity surfaceemitting lasers,” Proc. SPIE, vol. 7216, pp. 721619-1–721619-16,Feb. 2009.

[24] T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, “Highquality two-dimensional photonic crystal slab cavities,” Appl. Phys. Lett.,vol. 79, pp. 4289–4291, Oct. 2001.

[25] Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonicnanocavity in a two-dimensional photonic crystal,” Nature, vol. 425,pp. 944–947, Oct. 2003.

[26] S. Strauf, “Cavity QED: Lasing under strong coupling,” Nature Phys.,vol. 6, pp. 244–245, Feb. 2010.

[27] T. Yoshieet, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs,G. Rupper, et al., “Vacuum Rabi splitting with a single quantumdot in a photonic crystal nanocavity,” Nature, vol. 432, pp. 200–203,Nov. 2004.

[28] D. Englund, “Controlling the spontaneous emission rate of singlequantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett.,vol. 95, no. 1, pp. 013904-1–013904-4, 2005.

[29] W. H. Chang, W.-Y. Chen, H.-S. Chang, T.-P. Hsieh, J.-I. Chyi, andT.-M. Hsu, “Efficient single photon sources based on low densityquantum dots in photonic crystal nanocavities,” Phys. Rev. Lett., vol. 96,no. 11, pp. 117401-1–117401-4, 2006.

[30] M. Kaniber, A. Kress, A. Laucht, M. Bichler, R. Meyer, M.-C. Amann,et al., “Efficient spatial redistribution of quantum dot spontaneousemission from two dimensional photonic crystals,” Appl. Phys. Lett.,vol. 91, pp. 061106-1–061106-3, 2007.

[31] M. Kaniber, A. Laucht, T. Huerlimann, M. Bichler, R. Meyer,M.-C. Amann, et al., “Highly efficient single-photon emission fromsingle quantum dots within a two dimensional, photonic band-gap,”Phys. Rev. B, vol. 77, no. 7, pp. 073312-1–073312-4, 2008.

[32] O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus,et al., “Two-dimensional photonic band-gap defect mode laser,” Science,vol. 284, pp. 1819–1821, Jun. 1999.

[33] H. G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-K. Yang, J.-H. Baek,et al. “Electrically driven single-cell photonic crystal laser,” Science,vol. 305, pp. 1444–1447, Sep. 2004.

[34] H. Altug, D. Englund, and J. Vuckovic, “Ultrafast photonic crystalnanocavity laser,” Nature Phys., vol. 2, no. 7, pp. 484–488, 2006.

[35] D. Englund, A. Faraon, I. Fushman, N. Stoltz, P. Petroff, andJ. Vuckovic, “Controlling cavity reflectivity with a single quantum dot,”Nature, vol. 450, pp. 857–861, Dec. 2007.

[36] I. Fushmann, D. Englund, A. Faraon, N. Stoltz, P. Petroff, andJ. Vuckovic, “Controlled phase shifts with a single quantum dot,”Science, vol. 320, pp. 769–772, May 2008.

[37] G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer,“Vacuum Rabi splitting in semiconductors,” Nature Phys., vol. 2, no. 2,pp. 81–90, 2006.

[38] D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler,M. Kamp, et al., “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett.,vol. 98, no. 11, pp. 117402-1–117402-4, 2007.

[39] S. Mosor, J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova,H. M. Gibbs, et al., “Scanning a photonic crystal slab nanocav-ity by condensation of xenon,” Appl. Phys. Lett., vol. 87, no. 14,pp. 141105-1–141105-3, 2005.

[40] K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde,et al., “Quantum nature of a strongly coupled single quantum dot-cavitysystem,” Nature, vol. 445, pp. 896–899, Feb. 2007.

[41] A. Laucht, F. Hofbauer, N. Hauke, J. Angele, S. Stobbe, M. Kaniber,et al., “Electrical control of spontaneous emission and strongcoupling for a single quantum dot,” New J. Phys., vol. 11,pp. 023034-1–023034-12, Feb. 2009.

[42] C. Ciuti, G. Bastard, and I. Carusotto, “Quantum vacuum properties ofthe intersubband cavity polariton field,” Phys. Rev. B, vol. 72, no. 11,pp. 115303-1–115303-9, 2005.

[43] D. Dini, R. Köhler, A. Tredicucci, G. Biasiol, and L. Sorba, “Microcavitypolariton splitting of intersubband transitions,” Phys. Rev. Lett., vol. 90,no. 11, pp. 116401-1–116401-4, 2003.

[44] A. A. Anappara, S. De Liberato, A. Tredicucci, C. Ciuti, G. Bia-siol, L. Sorba, et al., “Signatures of the ultrastrong light-matter cou-pling regime,” Phys. Rev. B, vol. 79, no. 20, pp. 201303-1–201303-4,2009.

[45] G. Günter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, et al.,“Sub-cycle switch-on of ultrastrong light-matter interaction,” Nature,vol. 458, pp. 178–181, Mar. 2009.

[46] S. D. Liberato, C. Ciuti, and I. Carusotto, “Quantum vacuumradiation spectra from a semiconductor microcavity with a time-modulated vacuum Rabi frequency,” Phys. Rev. Lett., vol. 98, no. 10,pp. 103602-1–103602-4, 2007.

[47] M. Geiser, F. Castellano, G. Scalari, M. Beck, L. Nevou, and J. Faist,“Ultrastrong coupling regime and plasmon polaritons in parabolicsemiconductor quantum wells,” Phys. Rev. Lett., vol. 108, no. 10,pp. 106402-1–106402-5, 2012.

[48] T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke,M. J. Schwarz, et al., “Circuit quantum electrodynamics in theultrastrong-coupling regime,” Nature Phys., vol. 6, pp. 772–776,Jul. 2010.

[49] P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano,C. J. P. M. Harmans, et al., “Observation of the Bloch-Siegert shift ina qubit-oscillator system in the ultrastrong coupling regime,” Phys. Rev.Lett., vol. 105, no. 23, pp. 237001-1–237001-4, 2010.

[50] G. Scalari, C. Maissen, D. Turcinková, D. Hagenmüller,S. De Liberato, C. Ciuti, et al., “Ultrastrong coupling of the cyclotrontransition of a 2D electron gas to a THz metamaterial,” Science, vol. 335,pp. 1323–1326, Mar. 2012.

[51] Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liberato, C. Ciuti,P. Klang, et al., “Ultrastrong light-matter coupling regime with polaritondots,” Phys. Rev. Lett., vol. 105, no. 19, pp. 196402-1–196402-4, 2010.

[52] R. Stassi, A. Ridolfo, O. Di Stefano, M. J. Hartmann, andS. Savasta, “Spontaneous conversion from virtual to real photons inthe ultrastrong coupling regime,” Phys. Rev. Lett., vol. 110, no. 24,pp. 243601-1–243601-5, 2013.


[53] S. De Liberato, “Light-matter decoupling in the deep strong couplingregime: The breakdown of the Purcell effect,” arXiv: 1308.2812, 2013.

[54] A. H. Sadeghi, A. Naqavi, and S. Khorasani, “Interaction of quantumdot molecules with multi-mode radiation fields,” Sci. Iranica, vol. 17,pp. 59–70, Jun. 2010.

[55] E. Ahmadi, H. R. Chalabi, A. Arab, and S. Khorasani, “Cavity quan-tum electrodynamics in the ultrastrong coupling regime,” Sci. Iranica,vol. 18, no. 3, pp. 820–826, 2011.

[56] E. Ahmadi, H. R. Chalabi, A. Arab, and S. Khorasani, “Revisiting theJaynes-Cummings-Paul model in the limit of ultrastrong coupling,” Proc.SPIE, vol. 7946, pp. 79461W-1–79461W-6, Feb. 2011.

[57] A. Arab, “Cavity electro-dynamics in ultra-strong coupling,” M.S. thesis,School Electr. Eng., Sharif Univ. Technol., Tehran, Iran, 2011.

[58] A. Arab and S. Khorasani, “Anomalous evolution of quantum systemsin the ultrastrong coupling regime,” Proc. SPIE, vol. 8268, pp. 82681M-1–82681M-12, Jan. 2012.

[59] F. Karimi and S. Khorasani, “Optical modulation by conducting inter-faces,” IEEE J. Quantum Electron., vol. 49, no. 7, pp. 607–616,Jul. 2013.

[60] M. Alidoosty, “Behavior patterns analysis of cavity quantum electro-dynamics in complex systems for different coupling regimes,” M.S. the-sis, School Electr. Eng., Sharif Univ. Technol., Tehran, Iran, 2012.

[61] W. P. Schleich, D. M. Greenberger, D. H. Kobe, and M. O. Scully,“Schrödinger equation revisited,” Proc. Nat. Acad. Sci., vol. 110, no. 14,pp. 5374–5379, 2013.

[62] F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, “Atomiccoherent states in quantum optics,” Phys. Rev. A, vol. 6, no. 6,pp. 2211–2237, 1972.

[63] G. B. Lemos, R. M. Gomes, S. P. Walborn, P. H. Souto Ribeiro, andF. Toscano, “Experimental observation of quantum chaos in a beam oflight,” Nature Commun., vol. 3, no. 11, p. 1211, Nov. 2012.

[64] J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano,“Deep strong coupling regime of the Jaynes-Cummings model,” Phys.Rev. Lett., vol. 105, no. 26, pp. 263603-1–263603-4, 2010.

[65] M. Alidoosty Shahraki, S. Khorasani, and M. H. Aram, “Theoryand simulation of cavity quantum electro-dynamics in multi-partitequantum complex systems,” Appl. Phys. A, 2013, doi: 10.1007/s00339-013-8025-4.

[66] S. Prorok. (2013). Nanophotonics and integrated optics: Photonic crystalcavities. CST White Paper [Online]. Available: http://www.cst.com/

[67] H. Moya-Cessa, “Decoherence in atom-field interactions: A treatmentusing superoperator techniques,” Phys. Rep., vol. 432, no. 1, pp. 1–41,2006.

[68] S. J. Akhtarshenas, “Concurrence vectors in arbitrary multipartite quan-tum systems,” J. Phys. A, Math. General, vol. 38, no. 30, pp. 67–77,2005.

[69] K. de Greve, L. Yu, P. L. McMahon, J. S. Pelc, C. M. Natarajan,N.-Y. Kim, et al., “Quantum-dot spin–photon entanglement via fre-quency downconversion to telecom wavelength,” Nature, vol. 491,pp. 421–425, Nov. 2012.

[70] W. B. Gao, P. Fallahi, E. Togan, J. Miguel-Sanchez, and A. Imamoglu,“Observation of entanglement between a quantum dot spin and a singlephoton,” Nature, vol. 491, pp. 426–430, Nov. 2012.

Moslem Alidoosty was born in Kerman, Iran, on May 5, 1984. He received theB.Sc. and M.Sc. degrees in electrical engineering from Sistan and BaluchestanUniversity, Zahedan, and Sharif University of Technology, Tehran, Iran, in2008 and 2012, respectively. His current research interests include optoelec-tronics and quantum optics.

Sina Ataollah Khorasani (S’98–M’05–SM’09) was born in Tehran, Iran, onNovember 25, 1975. He received the B.Sc. degree in electrical engineeringfrom the Abadan Institute of Technology in 1995, and the M.Sc. and Ph.D.degrees in electrical engineering from the Sharif University of Technology,Tehran, in 1996 and 2001, respectively. He is currently a Tenured AssociateProfessor of electrical engineering with the School of Electrical Engineering,Sharif University of Technology. He has been with the School of Electrical andComputer Engineering, Georgia Institute of Technology, as a Post-doctoralfrom 2002 to 2004 and Research Fellow from 2010 to 2011. His activeresearch areas include quantum optics and photonics, and quantum electronics.

Mohammad Hasan Aram was born in Tehran, Iran, in 1987. He receivedthe B.Sc. and the M.Sc. degrees in electrical engineering from the SharifUniversity of Technology, Tehran, in 2009 and 2011, respectively. He iscurrently a Ph.D. student of electrical engineering with the Sharif University ofTechnology. His field of interest includes photonics, quantum optics, quantuminformation, and quantum computation.

1066 ieee journal of quantum electronics, …sina.sharif.edu/~khorasani/publications/64.pdf · 1066...

Documents