Appl Phys ADOI 10.1007/s00339-013-8025-4

Theory and simulation of cavity quantum electro-dynamicsin multi-partite quantum complex systems

Moslem Alidoosty Shahraki · Sina Khorasani ·Mohammad Hasan Aram

Received: 1 October 2013 / Accepted: 2 October 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract The cavity quantum electrodynamics of variouscomplex systems is here analyzed using a general versatilecode developed in this research. Such quantum multi-partitesystems normally consist of an arbitrary number of quan-tum dots in interaction with an arbitrary number of cavitymodes. As an example, a nine-partition system is simulatedunder different coupling regimes, consisting of eight emit-ters interacting with one cavity mode. Two-level emitters(e.g. quantum dots) are assumed to have an arrangement inthe form of a linear chain, defining the mutual dipole–dipoleinteractions. It was observed that plotting the system trajec-tory in the phase space reveals a chaotic behavior in the so-called ultrastrong-coupling regime. This result is mathemat-ically confirmed by detailed calculation of the Kolmogoroventropy, as a measure of chaotic behavior. In order to studythe computational complexity of our code, various multi-partite systems consisting of one to eight quantum dots ininteraction with one cavity mode were solved individually.Computation run times and the allocated memory for eachsystem were measured.

1 Introduction

Cavity quantum electrodynamics is a branch of quantum op-tics, which studies the interaction of quantum optical sys-tems and confined radiation in a cavity [1]. For the firsttime, Jaynes and Cummings and independently Paul pre-sented a model (JCPM) to describe the interaction of a two-level quantum emitting system with one cavity mode [2, 3].

M. Alidoosty Shahraki · S. Khorasani (B) · M.H. AramSchool of Electrical Engineering,Sharif University of Technology, P.O. Box 11365-9363, Tehran,Irane-mail: khorasani@sina.sharif.edu

Quantum entangled states have turned into a frontier topic ofinterest due to their various potential applications in quan-tum telecommunication, information, and computing.

Programmable multimode quantum networks [4] are gen-erated from various multimode entangled states, enablingthe capability of switching between different linear opticalnetworks. Simulation of multi-partite many body systemshas recently become feasible for the case of electronic wave-functions in solids where nearly accurate analytical meth-ods have been found [5]. However, precise numerical sim-ulation of multi-partite quantum systems is known to be anintractable problem, in particular when entanglement infor-mation is to be kept. For this purpose, simulation of quantumsystems composed of few partitions can be hardly done ona personal computer, while existing supercomputers hardlycan simulate as many as only 50 partitions [6]. The onlyknown solution so far has been to perform the simulationusing physical real simulators [6]. But usefulness of suchphysical systems is obviously limited to the class of prob-lems they can deal with.

It is the purpose of the present research to demonstratethe feasibility of full-scale simulation of multi-partite quan-tum systems on personal computers. We have developed anextensive code which is capable of solving arbitrarily com-plex quantum systems. We demonstrate the usefulness ofthis code by solving an example encompassing eight quan-tum dots interacting with one radiation mode. While thecode is generally capable of solving for larger numbers ofpartitions, memory restrictions on the laptop PC (intel Corei3 2.4 GHz, 4 GB RAM) used for this purpose have pre-vented us to explore more complex systems. We have sim-ulated the weak, strong, and ultrastrong-coupling regimesand confirm the onset of quantum chaos as a result of ultra-strong coupling, through calculation of the Kolmogorov en-tropy K2 [7]. It has to be added that the subject of quantum

M. Alidoosty Shahraki et al.

chaos in optical [8], microwave [9, 10], and nano-photonic[11] systems has been a matter of interest recently.

The theory has been based on an extension of the JCPMmodel [2, 3], and specification of all possible ket states. Theappropriate generalized JCPM must describe the interactionbetween emitters as dipole–dipole terms, as well as the in-teraction of emitters with multimode fields as field-dipoleterms. For this purpose, a special notation and formulationhas been invented [12], in conjunction with a major correc-tion to the JCPM model [13–15].

In [12], the model Hamiltonian is transformed to theHeisenberg interaction picture, and the subsequent Rabiequations were numerically solved in which RWA (Rotat-ing Wave Approximation) was vital. This transformation isnormally used in the context of quantum optics [1]. How-ever, subsequent studies [13–15] revealed that applicationof Heisenberg’s transformation to the interaction picture willproduce mathematically incorrect solutions, markedly underthe ultrastrong coupling. It is because the free-running solu-tions for field and atomic operators in ultrastrongly coupledsystems oscillate largely non-sinusoidal. Due to this fact, er-ror could be avoided by solving the system directly in theSchrödinger space without any approximation, rather thanthe Heisenberg interaction picture [15, 16]. In our most re-cent preprint [16] we analyzed a real quantum system mod-eled as a three-level system [17, 18], and another seven par-tition quantum system consisting of six quantum dots in in-teraction with one cavity mode [19].

In the present paper we are going to report our studyon the behavior of a complex nine-partition quantum sys-tem analyzed again under various coupling strengths. Here,emitters are assumed to have a linear chain arrangement,so that the dipole–dipole terms for non-neighboring dotsvanish. Under a similar two-dimensional configuration [6],many exciting applications are expected where a quantumdot transition is put in resonance with a photonic Diracpoint’s frequency [20]. We also study the computationalcomplexity of multi-partite systems versus the number ofpartitions for two algorithms based on full and sparse matri-ces. Counter-intuitively, usage of sparse matrices results inlonger computational times compared to full matrices, whileallocating the same order of memory.

2 Theory and algorithm

Required theory and algorithm are explained completely in[16], however, the basic definitions are explained here inbrief. The initial point of our methodology is to specify themost general time-dependent system ket state to representall possible entangled states, and the related generalized op-erating Hamiltonian. Then the time-dependent state of the

most general possible system will be given by

∣∣φ(t)

⟩ =∑

A,F

φ(A,F )|A〉|F 〉

|A〉 =k

⊗

n=1

∣∣∣∣

n

rn

⟩

=∣∣∣∣

1r1

⟩ ∣∣∣∣

2r2

⟩

. . .

∣∣∣∣

k

rk

⟩

1 ≤ rn ≤ Bn

|F 〉 =ω

⊗

ν=1

|fν〉 = |f1〉|f2〉 · · · |fω〉 0 ≤ fν ≤ N

(1)

Here, |A〉 is supposed to be the ket state of the different en-ergy levels of emitters, k is the total number of emitters,|F 〉 is the ket state of the cavity modes, fν is the numberof photons in νth cavity mode number, and ω is the totalnumber of cavity modes.

∣∣

nrn

⟩

expresses the eigenstate of thenth quantum dot residing at its rnth energy level. Also, thesubscript A refers to the different energy level states of lightemitters and Bn is the number of energy levels. F refersto the different cavity modes occupied with photons and N

is the maximum number of photons which possibly occu-pies a cavity mode. |φ(t)〉 is a superposition of all possi-ble states of the system, including atom and field states, andeach state has a time-dependent coefficient equal to φ(A,F )

[16].The generalized Hamiltonian H = H0 + Hr·E + Hr·r is

[16]

H0 =∑

n,i

Eni σ n

i +∑

ν

�Ωνa†ν aν

Hr·E =∑

n,i<j

(

γnij σni,j + γ ∗

nij σnj,i

)∑

ν

(

gnijν aν + g∗nijν a

†ν

)

Hr·r =∑

n<m,i<j

(

ηnij σni,j + η∗

nij σnj,i

)(

ηmij σmi,j + η∗

mij σmj,i

)

(2)

with H0 describing the system energy without interaction,Hr·E describing the light-emitter interactions, and Hr·r rep-resenting interactions between any possible pair of emitterssuch as dipole–dipole terms. Coefficients γnij are matrix el-ements of dipole operator of nth emitter. The strength of thedipole interaction between nth emitter and νth mode of cav-ity is given by gnijν with the transition ith and j th energylevels. Coefficients ηnij are proportional to the strength ofthe dipole generated while another emitter undergoes a tran-sition between ith and j th levels. En

i indicates the ith energyof the nth emitter av and aν are the field creation and annihi-lation operators, which, respectively, increase and decreasethe number of photons in the vth cavity mode by one, σ l

s,k isladder operator which makes the lth emitter to switch fromkth to sth level.

Theory and simulation of cavity quantum electro-dynamics in multi-partite quantum complex systems

Fig. 1 Assumed light emitting systems arrangement

Now, the governing Schrödinger equation is

∂

∂t

∣∣φ(t)

⟩ = − i

�H

∣∣φ(t)

⟩

(3)

which can be recast into the form with the closed-form so-lution

d

dt

{

Φ(t)}

N×1 = [M]N×N

{

Φ(t)}

N×1 (4)

{

Φ(t)} = exp

([M]t){Φ(0)}

(5)

In our code both sides of (3) are compared and coefficientswill be arranged as elements of an N × N square matrix[M]N×N , where N is the total number of possible states.Then (3) is rewritten as (4), which admits an analytical so-lution (5) subject to the initial conditions {Φ(0)}. We havealso developed an efficient, stable, and accurate method toevaluate (5) [13, 16, 19]. Then, it would be possible to evalu-ate the probabilities and expectation values accurately underdifferent coupling regimes.

3 Numerical analysis results

Here we study a nine-partition quantum optical system. Firstthe system is characterized, then the state coefficients andsubsequent required values to study the system behavior areevaluated precisely. The values are probabilities and expec-tation values of atomic and ladder operators.

3.1 Characterization

It is supposed that this nine-partition system consists of eightidentical light emitters (such as quantum dots) which arelimited to two different energy level states. We take on atransition energy of 1 eV between the ground and excitedstate. Light emitters are assumed to have a linear chain ar-rangement as Fig. 1, where dipole–dipole interactions are al-lowed only between the nearest neighbors. We actually havetested the validity of this approximation by taking a secondneighbor’s coupling strength to be an order of magnitudeless than that the first neighbor’s, and found that the resultswere essentially indistinguishable. This interaction has thetypical magnitude of 5 meV.

It is furthermore supposed that transition dipole momentin these quantum dots is 96 Debye, in accordance with thetypical values calculated elsewhere [17, 18]. We let one

Table 1 Rabi frequencies

E0 ( kVcm ) Ge,g( Rad

s )Ge,g

ωλ

20 6.0676 × 1012 0.004

200 6.0676 × 1013 0.04

2000 6.0676 × 1014 0.4

resonant cavity mode with frequency ωλ and the Rabi fre-quency Ge,g exist given by

ωλ = 1

�(Ei − Eg) = 1.5177 × 1015 Rad/sec (6)

Ge.g = 1

�(E0E) · 〈ψe|eR|φg〉 (7)

Here, |ψe|eR|φg〉 is the transition dipole moment, E0E isthe applied electrical field vector, and � is reduced Plankconstant. The electrical field E0 determines the Rabi fre-quencies, whereas by comparing the Rabi frequencies withthe optical frequency ωλ, different coupling regimes aresimulated. According to (7), Rabi frequencies are given inTable 1, for weakly, strongly, and ultrastrongly coupled sys-tems, based on typical field strengths.

Based on the data in Table 1, for the weakest applied elec-tric field with Ge,g = 0.004ωλ, the system is in the weak-coupling regime. In a stronger electric field with Ge,g =0.04ωλ the coupling regime is strong. The strongest elec-tric field provides Ge,g = 0.4ωλ, which takes the couplingregime into the ultrastrong domain.

In this system, we assume the maximum possible numberof occupying photons in the cavity mode to be eight. There-fore the general time-dependent state of the system accord-ing to the (1) is

∣∣φ(t)

⟩ =∑

r1···7,r8=e,g

8∑

f =1

φ(A,f )|A〉|f 〉

|A〉 =8

⊗

n=1

∣∣∣∣

n

rn

⟩

=∣∣∣∣

1r1

⟩ ∣∣∣∣

2r2

⟩

. . .

∣∣∣∣

8r8

⟩

1 ≤ rn ≤ 2

|f 〉 = |f1〉 0 ≤ f1 ≤ 8

(8)

in which r1, . . . , r8 are the quantum dots. The related Hamil-tonian of this system according to (2) will be

H0 =8

∑

n=1,i=e,g

Eni σ n

i + �Ωa†a

M. Alidoosty Shahraki et al.

Hr·r =∑

n<m,i<j

(

ηnij σni,j + η∗

nij σnj,i

)(

ηmij σmi,j + η∗

mij σmj,i

)

=n=7m=8

∑

n=1<m=2,i=g<j=e

(

ηnij σni,j + η∗

nij σnj,i

)(

ηmij σmi,j

+ η∗mij σ

mj,i

)

(9)

Hr·E =8

∑

n=1,i=g<j=e

(

γnij σni,j + γ ∗

nij σnj,i

)(

gnij a + g∗nij a

†)

A coherent field has been considered as initial state. Theextended initial condition, based on the existing relations foran arbitrary quantum optic system [16] the initial coherentfield for this system, with λ being the coherence number (orλ2 being the average photon number), is

∣∣φ(0)

⟩ = 1√28R

8∑

n=0

√

λn

n! e−λ|n〉

×∑

r1...r8=g,e

|r1, r2, r3, r4, r5, r6, r7, r8〉

R =8

∑

n=0

∣∣∣∣

√

λn

n! e−λ

∣∣∣∣

2

(10)

3.2 Presence probability with coherent initial state

According to (1), the presence probability of an arbitrarylight emitting system such as l, being at a typical energylevel such as k, is simply [14]

P =∑

A−{rl}=k

N∑

f1,f2,...,fν=0

∣∣φ(r1, r2, . . . , rl→k, rn, f1, . . . fν)

∣∣2

(11)

Subsequently, the presence probability of a quantum dot be-ing at its ground and excited energy level was measured ac-cording to

P =∑

A−{r1}=g

8∑

f1=0

∣∣φ(r1→g , r2, . . . , r8, f1)

∣∣2

P =∑

A−{r1}=e

8∑

f1=0

∣∣φ(r1→e , r2, . . . , r8, f1)

∣∣2

(12)

These probabilities for all eight quantum dots were plotted.Figure 2 presents the presence probability of first quantumdot for weak, strong, and ultrastrong coupling. As the cou-pling constant increases, anonymous oscillations increaseand the chaotic behavior of ultrastrong coupling can be seenagain in this system as in previous works [16].

3.3 Field operators in different coupling regimes

The behavior of field annihilation and creation operators iscalculated in this work, too. The expectation value of theannihilation operator is obtained as [16]

⟨

φ(t)∣∣aν

∣∣φ(t)

⟩ =∑

A,F

√

fνφ∗(A,fν − 1)φ(A,fν)

=∑

r1,2,...,n−1,rn=g,e

N∑

f1,f2,...,fv,fm=0

√

fνφ∗(r1, r2, . . . , rn, f1,

f2, . . . , fν − 1, . . . , fm)

× φ(r1, r2, . . . , rn, f1, f2, . . . , fν, . . . , fm)

(13)

So, the expectation value of annihilation operator is foundas

⟨

φ(t)∣∣aν

∣∣φ(t)

⟩ =∑

r1···7,r8=g,e

8∑

f1=0

√

f1φ∗(r1, . . . , r8, f1 − 1)

× φ(r1, . . . , r8, f1) (14)

Phase space plots and flat phase plots of the real and imag-inary values of the expectation value as functions of nor-malized time are shown in Fig. 3, respectively, for weakly,strongly, and ultrastrongly coupled systems.

As is observed, by increasing the coupling constant andentering into ultrastrong regime the system behaves verychaotic, marking a disordered behavior. Also, the corre-sponding phase changes nonlinearly. This behavior was alsoseen in complex ultrastrongly coupled multipartite systemsanalyzed before [13–16, 19], with the difference that inthe previous works, the corresponding phase would un-dergo abrupt random-like step changes. But as is seen notonly by increasing the partitions the behavior is not thesame as before, but also it is very disordered in shortertime durations and seems to be constant in long dura-tions.

3.4 Expectation value of the atomic ladder operator

The behavior of ladder operator has been also computed,with the expectation value of annihilation operator in an ar-bitrary quantum optic system being [16]

Theory and simulation of cavity quantum electro-dynamics in multi-partite quantum complex systems

Fig. 2 Presence probability ofthe first quantum dot plotted forboth ground and excited states.First, second and third rows arein weak-coupling regime,strong-coupling regime andultrastrong-coupling regime,respectively

⟨

φ(t)∣∣σ l

s,k

∣∣φ(t)

⟩

=∑

A−{rl},Fφ∗(Arl→s ,F )〈F |〈Arl→s |φ(Arl→k,F )

× |Arl→s〉|F 〉=

∑

A−{rl},Fφ∗(Arl→s ,F ) × φ(Arl→k,F )

=∑

A−{rl}

N∑

f1,f2,...,fm=0

φ∗(r1, r2, rl → s, . . . , rn, f1, f2,

. . . , fm)

× φ(r1, r2, rl → k, . . . , rn, f1, f2, . . . , fm)

(15)

Here, only the second dot as an example is presented (allother dots have the same typical behavior). So by using (15)

the expectation value of ladder operator for the transitionof every quantum dot from excited energy level to groundenergy level will be

⟨

φ(t)∣∣σ 2

g,e

∣∣φ(t)

⟩

=∑

r1,3,4,5,6,7,8=g,e

8∑

f1=0

φ∗(r1, r2 → g, r3, . . . , r8, f1)

× φ(r1, r2 → e, r3, . . . , r8, f1) (16)

Three-dimensional and phase plots of the real and imag-inary values of the measured expectation value as functionsof normalized time duration in Fig. 4, show chaotic behav-ior and nonlinearity in ultrastrong-coupling regime is ob-served. In this system, disordered behavior increases signif-icantly and the same anonymous behavior in phase spaceand phase plots even seems to begin from strong-coupling

M. Alidoosty Shahraki et al.

Fig. 3 Phase space and phaseplots of expectation value ofannihilation operator. First,second and third rows are inweak-coupling regime,strong-coupling regime andultrastrong-coupling regime,respectively

regime. Moreover, the amplitude of the real and imaginaryparts decreases so much.

It is not very difficult to mathematically show thatthe intensity of chaos indeed increases with the couplingstrength. For this purpose, we calculate the Kolmogoroventropy K2 defined in [7] for the annihilator and atomicladder expectation complex values as functions of time,as show in Fig. 5. The K2 entropy is always positive;a vanishing K2 corresponds to a completely non-chaoticbehavior, while it approaches infinity for fully randomsignals, i.e. white noise. In general, any finite non-zeroK2 entropy is a sufficient condition of chaotic behav-ior [7].

4 Computation of complexity

According to (1), the number of possible states of a quantumoptical system, here denoted by M , equals

M = (N + 1)Cavitym × (energylev)QDno (17)

where N is the maximum number of occupying photons incavity modes, with the number of Cavitym. QDno is num-ber of light emitting systems that each one has energylevnumber of energy levels. Thus, M is a good measure of thenumerical dimension of the problem under consideration.

So it is obvious that the computational complexity willbe a strong function of the number of system partitions ac-cording to (17). Since the number of possible states in an

Theory and simulation of cavity quantum electro-dynamics in multi-partite quantum complex systems

Fig. 4 Phase space and phaseplots of expectation value of theatomic ladder operator, first,second, and third rows are inweak-coupling regime,strong-coupling regime andultrastrong-coupling regime,respectively

Fig. 5 Increasing K2 entropy versus coupling regime shows the de-pendence of chaotic behavior on coupling strength

analyzing system would be so large, it was inevitable to pro-vide an extensive computer code to tackle such systems with

an arbitrary number of states. Therefore, a very comprehen-sive MATLAB software has been developed to achieve thisgoal.

In the software we firstly utilized full matrices to pro-vide the initial desired matrix and compute the state co-efficient matrix. However, memory requirements naturallylimit the matrix dimensions, and hence the selection of asystem complexity was limited. For example, considerationof a system larger than ten partitions (9 QDs in interactionwith 1 Cavitym with 7 photons) was not possible on the PCwe were using. Hence we revised the codes with sparse ma-trices. The sparse matrix squeezes out any zero element ofa full matrix, so that it made possibility to consider a sys-tem as large as 228 possible states for example a 26 partitionsystem (25 QDs in interaction with 1 Cavitym with 7 pho-tons). However, the necessity of eigenvalue computation inthe process of state measurement, the sparse matrices allo-cate more amount of RAM, resulting in increased run time

M. Alidoosty Shahraki et al.

Fig. 6 Allocated memory and run time of software processing. Firstplot is allocated memory in both full and sparse matrix utilization. Thesecond plot is the run time for full matrix application and sparse matrix,respectively

process. So, as long as the eigenvalue extraction is not satis-factorily efficient, use of sparse matrices counter-intuitivelywould increase the computational complexity.

In order to study the trend of complexity and its relationto the number of system partitions, various systems havingtwo to nine partitions were solved, through the measurementof their state coefficient matrix. Run time and memory allo-cation for analyzed selected systems are plotted in Fig. 6to illustrate the computational complexity of our algorithmsand code. According to the results obtained, by increasingthe number of partitions both the required RAM and thecomputational burden surge exponentially. We estimate thatthe overall efficiency of the algorithm based on full matricesis proportional of 9.6M with M defined in (17), whereas forsparse matrices one would expect a computational burdenproportional to 17.8M .

5 Conclusions

In this paper, the cavity quantum electrodynamics of a com-plex system with nine partitions has been analyzed. The fun-damental methodology to tackle such systems theoreticallywas mentioned. In the ultrastrong-coupling regimes, similarto our previous works, a remarkable chaotic behavior hasbeen again observed. However, it appears that the chaoticbehavior is pronounced at larger number of partitions. Also,phase plots illustrate strong disorder along with a differ-ent behavior not seen earlier, which showed abrupt stepwisephase changes. Then, we repeated our simulation for variouscomplex systems, while monitoring the overall run time andallocated memory. Since the coefficients matrix is largelysparse, we would normally anticipate that sparse algorithmswould cause less computational burden. Strikingly, it wasobserved that it was not the case, and we would relate thiscounter-intuitive conclusion to the inefficiency of existingeigenvalue extraction methods for sparse matrices.

Acknowledgements This work was supported in part by Iranian Na-tional Science Foundation (INSF) under Grant 89001329.

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