Cavity quantum electrodynamics of multipartite systems

Cavity quantum electrodynamics of multipartite

systems

Moslem Alidoosty Shahraki1 and Sina Khorasani1,*

1School of Electrical Engineering, Sharif University of Technology, P. O.Box 11365-9363, Tehran, Iran *Corresponding author: [email protected]

Cavity quantum electrodynamics of multipartite systems is studied in depth, which consist of an arbitrary number

of emitters in interaction with an arbitrary number of cavity modes. The governing model is obtained by taking the

full field-dipole and dipole-dipole interactions into account, and is solved in the Schrödinger picture without

assumption of any further approximation. An extensive code is developed which is able to accurately solve the

system and track its evolution with high precision in time, while maintaining sufficient degrees of arbitrariness in

setting up the initial conditions and interacting partitions. Using this code, we have been able to numerically

evaluate various parameters such as probabilities, expectation values (of field and atomic operators), as well as the

concurrence as the most rigorously defined measure of entanglement of quantum systems. We present and discuss

several examples including a seven-partition system consisting of six quantum dots interacting with one cavity

mode. We observe for the first time that the behavior of quantum systems under ultrastrong coupling is

significantly different than the weakly and strongly coupled systems, marked by onset of a chaos and abrupt phase

changes.

1. INTRODUCTION

Without doubt, Cavity Quantum Electrodynamics (CQED) is one of the frontiers of modern science, where its applications

are rapidly entering the realm of engineering fields. With the advent of quantum computing, CQED remains 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 complicated analytical and simulation tools

capable of dealing with multipartite systems without losing accuracy. As the number of partitions increase, the difficulty

in treatment of multipartite systems unfolds in two aspects: (a) how to maintain accuracy while increasing the system

dimensionality, and (b) how to write down equations using proper notations and mathematical expression without causing

confusion and/or ambuigity. This is while we have to add the computational complexity of solving algorithm unaddressed.

The first successful theoretical understanding of CQED was made by Jaynes and Cummings in 1963 [1,2] and

independently by Paul [3]. The beauty and effectiveness of Jaynes-Cummings-Paul Model (JCPM) is in the fact that it

gives out an explicit solution for resonantly interacting two-level emitters and one radiation mode, while non-resonant

solutions are offered as inifinite series with exactly solvable time-dependencies. Soon after, JCPM proved its usefulness in

description of then-poorly understood quantum phenomena, such as spontaneous emission [4], and also prediction of

collapse and revival of wavefunctions [5], which was not observed in expreiments till the next three decades [6]. Further

development in experimental techniques and preparation of Rydberg atoms having very large transition dipole moments

allowed various interaction regimes to be studied with regard to the magnitude of coupling constant, which is better

known as Rabi frequency [7,8]. Alongside its clear and very well-known physical interpretation as the beat frequency, the

Rabi frequency may be also simply regarded as a measure of emitter-field interaction strength, which is an increasing

function of the dipole moment and field intensity.

According to the strength of the coupling constant, a CQED system may fall in three regimes: weakly, strongly, or

ultrastrongly coupled. If the Rabi frequency is less than the decay rate of excited states in the cavity then the interaction is

usually weak, which happens to address most of the occurring CQED systems. Weakly coupled systems are responsible for

a number of well-known phenomena such as enhanced or suppressed spontaneous radition, which have found applications

in modern light emitting devices such as semiconductor lasers [9,10].

If we continue to increase the coupling constant, or Rabi frequency, to such level that it would exceed one-fourth of the

difference of atom and field decay rates [11], then we enter the strong coupling regime. Under strong coupling, the

eignstates would be no longer degenerate and recombine to form two separate state pairs [12,13], having an energy

difference given by the Rabi frequency. If the emitting system is a quantum dot or well, then these new quantum mixed

states obtained from photon-exciton interaction are sometimes viewed as new quasi-particles referred to as exiton-

polaritons.

One of the important implications of strongly coupled systems is the possibility of solid-state quantum information

processesing [14]. Strong coupling causes the emitter and field not to be in complete resonance because of the mode

splitting which happens at the removal of degenerate modes. This behavior is known as anti-crossing, and is accompanied

by an energy gap which is a very clear experimental signature of strong coupling. As a consequence, novel quantum

phenomena including anti-bunching are obtained which form the basis of single-photon emitters [15], quantum encryption

[16], quantum repeaters [17,18], and quantum computation [19]. Strong coupling in a cavity may be reached by increasing

confinement times and thus quality factors, while decreasing the effective mode volume [20]. At optical frequencies,

semiconductor cavities [21] are able to match these criteria. Crossing energy gaps as large as 140eV have been observed.

Micro-disk [22] cavities have demonstrated successful operation of ultra low threshold lasers, and polariton laser emissions

are observed from GaN microcavities. Photonic crystal cavities are also able to combine excellent confinement and tight

mode volumes to enter the strong coupling regime [24-26]. Collapse and revival of wavefunctions in photonic crystal

nanocavities has been reported [27].

Among the other state-of-the-art problems in CQED systems, the control of detuning frequency can be mentioned,

which is the absolute difference between the emitter and field frequencies. High-fidelity single photon sources [28-31], high

bandwidth low-threshold lasers [32-34], and single quantum dot devices such as mirrors [35] and phase shifters [36] are

highly dependent on the possibility of control on detuning. Among these, various methods such as cryogenic lattice

temperature control [21,37,38], condensation at ultralow temperatures [39,40] and electrical control [41] may be

mentioned.

The next interaction regime is the ultrastrong coupling, in which the Rabi frequency is typically comparable or even

larger than the spontaneous emission decay rate. As a matter of fact, the interaction energy obtained by muliplication of

Planck‟s constant and Rabi frequency has a magnitude in the same order of the transition energy [42], which result in

improved excited and ground state properties such as non-adiabatic CQED phenomena [42]. Most ultrastrongly coupled

systems depend on the range of the radiation spectrm occure in either of the two solid state systems. At the optical

frequencies, intersubband transitions of semiconductor cavities placed in doped potential wells [43-47] form the physicaly

ultrastrong interaction. This is while at the microwave frequencies, superconductor resonator circuits supercooled to milli-

Kelvin temperatures in resonance with two-level emitters formed from Josephson junctions may interact ultrastrongly

[48,49]. More recently at the Terahertz frequencies, a third ultrastrongly coupled system has been identified [50], which is

obtained from the interaction of magnetic cyclotron resonances of a high-mobility two-dimensional electron gas in an

amplifying medium. Since the cyclotron frequency, as the transition frequency, is very well controllable with a

perpendicular magnetic field, the transition dipoles are easily controlled and may be increased to extremely high values.

Rabi frequencies as large as 58% of the transition frequencies have been so far measured [50]. Also, metal-dielectric-metal

microcavities along with quantum wells form ideal systems for generation of cavity polaritons at the Terahertz spectrum.

Metal cavities encompass strongly confined modes, which may be placed at resonance with the intersubband transitions of

quantum wells. Ultrastrong coupling with Rabi frequencies exceeding 50% of the transition frequency has been

demonstrated [51].

In the simple JCPM, which formulates the quantum dynamics of a two-level atom in a radiation reservoir, the

Hamiltonian is transformed into the so-called Heisenberg‟s interaction picture [1]. Under such circumstances, the Bosonic

field creation and annihilation operators, obey simple first-order differential equations with solutions varying in time as

. These free-running solutions for field operators clearly oscillate completely sinusoidal in time. The interaction is

then expressed as the summation of fast (sum of field and transition frequencies) and slow (difference of field and

transition frequencies) components. Normally, the fast component is associated with extremely high frequencies compared

to the slow components, and hence are ignored. This approximation, in fact, discards the energy-violating terms and

referred to as the Rotating Wave Approximation (RWA) [1].

The Heisenberg‟s interaction picture in JCPM results in a time-dependent interaction Hamiltonian [51] and solvable

state kets, while the original Schrödinger picture [52] has a time-independent Hamiltonian. The mathematical beauty of

Heisenberg‟s interaction picture in JCPM is lied within the fact that every two-level emitter is eventually treated with the

same algebra as a Fermionic particle with half-integer spin [53,54]. Therefore, Pauli matrices normally enter the

mathematical formulation of atomic transitions. It is also possible to extend the JCPM for three-level atoms, which has

been reported elsewhere [55]. Other different aspects of the theory of JCPM have been thoroughly studied and published in

the past [56-59], including the coherent [60], squeezed states [61], increased number of energy levels [62] and cavity

radiation modes [63].

More recently, the entangled quantum states have become a center of interest because of their primary role in

quantum communications and information [64-66]. For instance, a new paper [67] discusses how programmable quantum

networks could be realized and controlled using these highly entangled states constructed in multipartite quantum

systems. Multipartite systems having more than three partitions also appear in other complex systems obtained from

several atoms [68], multiple phonon transitions [69,70], as well as intensity-dependent entangled systems [71]. To this list

of solvable models, we may add a three-level atom interacting with one [72], or two radiation modes [73].

Recently, we have considered the most general CQED system [74] comprising an arbitrary number of emitters and

radiation modes subject to an arbitrary initial state. Useful mathematical formulation and analysis of such a system is

highly contingent on a different and extended notation of atomic and field states and operators, which we have constructed

therein. We furthermore have allowed field-dipole and dipole-dipole interactions to exist. Then the model Hamiltonian is

transformed to the Heisenberg‟s interaction picture under RWA, and the subsequent Rabi equations are numerically

solved. This formulation was ultimately incomplete, since it is now very well known that RWA should not be applied to the

ultastrongly coupled systems.

In our next researches [75-78] it has been furthermore found that application of Heisenberg‟s transformation to the

3

interaction picture results in mathematically incorrect solutions, most visibly under the ultrastrong coupling. The reason

is that free-running solutions for field operators in ultrastrongly coupled systems are no longer correct, and they oscillate

largely non-sinsusoidal. Because of this, the usefulness of Heisenberg‟s transformation vanishes, and we may conclude

that ultrastrongly coupled systems may be correctly studied, regardless of RWA, only in the Schrödinger‟s picture.

It is the purpose of this paper to construct and accurately solve a self-consistent and most general theoretical picture

of multipartite CQED systems without RWA and/or Heisenberg‟s transformation, while being applicable to ultrastrongly

coupled systems. The explicit solution is greatly simplified and accurately evaluated using algebraic matrix exponential

techniques, as just reported in a recent publication [78]. This technique allows rapid and accurate evaluation of state kets

in time without relying on any numerical integration. For treatment of partitions (or more accurately, particles) having

spin, and in particular Fermions with half-integer spin, we may note that the corresponding spinors are actually special

cases of two-partite entangled systems obtained by outer product of two scalar Fermions. Hence, the present formulation is

equally capable of dealing with spin at the expense of a two-fold increase in the number of scalar partitions. As a result, we

have been able for the first time to investigate ultrastrongly coupled multipartite systems not been studied so far [79].

These include an ultrastrongly coupled integrated waveguide structure realized in compound semiconductor quantum

wells [80], which is modeled as a two-partition system consisting of a three-state -emitter and one radiation modes. The

other example is a seven-partition system consisting of six identical two-level quantum dots and one radiation mode,

having a maximum photon occupany number of twenty-four.

By plotting various probabilities and expectation values of field and atomic operators, we can find that ultrastrongly

coupled systems are marked by onset of a chaotic behavior in phase space. Earlier this year [78] we had reported

anomalous and largely nonlinear variations for phases of field operators. Our present study for the first time sets up a

rigorous and novel method to trace the evolution of multi-partite systems in phase space, in which such chaotic behavior

are easily detectable. Our computer software code is theoretically capable of dealing with any multipartite CQED system,

and is able to self-generate an internal subroutine for exact calculation of concurrence. This parameter represents the

overall degree of entanglement in a multipartite system.

2. MATHEMATICS AND ALGORITHM To analyze the general behavior in CQED of complex multi-partite systems, initially the coefficients matrix of the most

general system has been computed and numerically measured. Such systems normally consist of an arbitrary number of

emitters (usually quantum dots) in interaction with an arbitrary number of cavity modes. For this purpose, the general

time-dependent state of the most general possible system has been rigorously specified and solved exactly in time-domain

using an explicit analytical solution presented in this section. As it will be shown, providing initial conditions as one of the

parts of the solution was vital. Fock and coherent initial conditions were considered in this article, so the most general

equation to provide such coherent initial condition was extracted. Required equations to measure the presence probability

of the system at different states were presented. Expectation values of field and atomic operators as well as the

expectation value of commutator of atomic ladder operators were also extracted. Finally an extensive high-level MATLAB

code is developed which sets up the initial conditions for any arbitrary complex system and evaluates mentioned

parameters including probabilities, expectation values of field and atomic operators, the commutator of atomic ladder

operators, as well as concurrence as the most general measure of entanglement of multipartite systems.

2.A. Coefficients matrix The aim of the provided MATLAB code in this part is to solve the Schrödinger equation and the coefficients matrix. Here, | ( )⟩ is the general state of system, is the generalized JCPM Hamiltonian presented in [74] and is the reduced Plank

constant.

| ( )⟩

| ( )⟩ (1)

Eq. (1) is solved in Schrödinger space in which the ket states of the system are time-dependent while operators are not.

Due to the reasons discussed earlier [75-78], Heisenberg‟s transformation has not been used.

We suppose that | ⟩ is the ket state of the different energy levels of emitters, is the total number of emitters, | ⟩ is

the ket state of the cavity modes, is the number of photons in ν –th cavity mode number, and ω is the total number of

cavity modes. Then the general time-dependent state of the most general possible system will be given by

| ( )⟩ ∑ ( )| ⟩| ⟩

| ⟩

|

⟩ |

⟩ |

⟩ |

⟩

| ⟩

| ⟩ | ⟩| ⟩ | ⟩ (2)

So according to the (2), various states are formed by the different states of being quantum dots at differnet energy levels

multiplied by different photon number states in each cavity mode. If is the total number of cavity modes, is the

number of energy levels, and N is the number of cavity modes then the expansion coefficients in (2) are given as

( ) ∑ ∑ ( )

(3)

The generalized JCPM Hamiltonian consists of describing the system energy without interaction, standing for

light-emmiter interactions, and representing interactions between any possible pair of emitters such as dipole-dipole

terms [74]:

+ (4)

∑

∑

(5)

∑ .

/ ∑ .

/ (6)

∑ .

/ .

/ (7)

Coefficients are matrix elements of dipole operator of -th emitter. The strength of the dipole interaction between -th

emitter and -th mode of cavity is given by with the transition -th and -th energy levels. Coefficients are

proportional to the strength of the dipole generated while another emitter undergoes a transition between -th and -th

levels. indicates the -th energy of the -th emitter. We furthermore have

| ⟩ √ | ⟩ (8)

| ⟩ √ | ⟩ (9) | ( )⟩

∑ ( ) | ⟩| ⟩ √ ∑ ( )| ⟩| ⟩

(10) | ( )⟩ ∑ ( ) | ⟩| ⟩ √ ∑ ( )| ⟩| ⟩

(11)

| ( )⟩

∑ ( ) | ⟩| ⟩ ∑ ( )| ⟩| ⟩ * + (12)

( ) | ( )⟩ (

) ∑ ( )| ⟩| ⟩ ∑ ( )| ⟩| ⟩ * + (13)

where and are the field creation and annihilation operators which respectively increase and decrease the number of

existing photons within the -th cavity mode by one, is atomic ladder operator which makes the -th emitter to switch

from -th to -th level, and ( ) is the adjoint of the

. Other parameters are those introduced before [78].

We evaluate the effects of each term of separately and the final results will be added together. By operation of (5),

we get

| ( )⟩ {∑

∑

}| ( )⟩ ∑

∑ ( )| ⟩| ⟩ ∑ √ √ ∑ ( )| ⟩| ⟩

(14) By the operation of (6), we get

| ( )⟩ 2∑ .

/ ∑ .

/ 3 | ( )⟩

∑ .

)| ( )⟩ / ∑ .

)| ( )⟩/

∑ ( √ ∑ ( )| ⟩| ⟩ * + ) ∑ . √ ∑ ( )| ⟩| ⟩ * + /

∑ . √ ∑ ( )| ⟩| ⟩ * + / ∑ .

√ ∑ ( )| ⟩| ⟩ * + /

(15)

By the operation of (7), we get

| ( )⟩ ∑ .

/ .

/ | ( )⟩ ∑ .

/ | ( )⟩

∑ .

| ( )⟩/

∑ ( ∑ ( )| ⟩| ⟩ * + ) ∑ .

∑ ( )| ⟩| ⟩ * + /

∑ .

∑ ( )| ⟩| ⟩ * + / ∑ .

∑ ( )| ⟩| ⟩ * + /

(16) And finally by adding up (14,15,16), we can construct the Schrödinger equation as

| ( )⟩

( | ( )⟩ | ( )⟩ | ( )⟩) (17)

The method of solving (17) is that for each state, both sides are compared and the coefficients are arranged as elements of

a square matrix [M]. If N is the number of different possible states, we get the system of equations

* ( )+ , - * ( )+ (18)

in which * ( )+ is the vector of unknown coefficients. We suppose that * ( )+ is the initial condition vector

5

* ( )+ , - * ( )+ (19)

To evaluate (19), [M] matrix is diagonalized into a diagonal matrix , - [ ] of eigenvalues using the diagonalizer , -

which is found from eigenvectors of [M] * ( )+ , - * ( )+ , - , - , - * ( )+ , -[ ], - * ( )+

(20)

The solution (20) is explicit and can be accurately evaluated regardless of the time . It also excludes the need of matrix

exponentiation and is thus numerically stable. We also note that * ( )+ can take on any initial such as Fock or coherent

[60] initial states. For instance, a coherent state with λ being the average number of photons is

| ( )⟩ ∑ √

| ⟩ (21)

Because maximum number of photons in the cavity mode is limited to a number such as m, so in order to ensure the

normalization of the states (21) we rewrite (21) as

| ( )⟩ √

∑ |√

|

∑ √

| ⟩ | ⟩ (22)

And for the most general multipartite system consisting of an arbitrary number of modes and emitters, the initial coherent

states will be

| ( )⟩ ∑√

∑ |√

|

√

(

√

∑ |√

|

√

| ⟩ | ⟩ (23)

2.B. Probabilities of presence at states According to (3), the presence probability of an arbitrary light emitting system such as , being in at arbitrary energy level

such as is simply

∑ ∑ | ( )|

* + (24)

2.C. Expectation values of field operators By the use of (8,9,10,11), the expectation values of annihilation and creation operators for the most general possible

complex system are found as

⟨ ( )| | ( )⟩ ∑ √ ( ) ( ) ∑ ∑ √

( )

( ) (25)

⟨ ( )| | ( )⟩ ∑ √ ( ) ( ) ∑ ∑ √ ( )

( ) (26)

2.D. Expectation values of ladder operators

By the use of (12,13) the expectation values of the atomic transition operators and their Hermitian adjoint are found as

⟨ ( )| | ( )⟩ ∑ ( )| ⟩| ⟩ * + ( )| ⟩| ⟩ ∑ ( ) * + ( )

∑ ∑ ( )

* + ( ) (27)

⟨ ( )|( ) | ( )⟩ ∑ ( )| ⟩| ⟩ * + ( )| ⟩| ⟩ ∑ ( ) * + ( )

∑ ∑ ( )

* + ( ) (28)

2.E. Expectation value of the ladder commutator

The commutator of the atomic ladder operators is the commutation of atomic transition operator and its Hermitian adjoint,

denoted by [

( ) ], with the expectation value found as

⟨ ( )|[ (

) ] | ( )⟩ ⟨ ( )|( (

) ( )

) | ( )⟩ ⟨ ( )|( (

) ) | ( )⟩ ⟨ ( )|(( )

) | ( )⟩

(29) In order to simplify (29), the following orthogonality equations are also needed [74]

(

) (

)

(30)

( )

( )

(31)

Now by the use of (30,31), we may write down

⟨ ( )|[ (

) ] | ( )⟩ ∑ ( ) * + ( ) ∑ ( ) * + ( ) ∑ | ( )|

* +

∑ | ( )|

* + (32)

3. ANALYSIS AND NUMERICAL RESULTS

Here, we present detailed analysis of two different quantum optical systems: a three-level quantum well and a

multipartite quantum optical system with seven partitions. All extracted equations in the previous section by the

utilization of our provided code are executed. Entaglement is also analyzed in both systems.

3.A. CQED in a real optoelectronic system In this section, we present the simulation and analysis of the CQED of a real complex system consisting of a three level

light emitting system interacting with a cavity mode. The emitter is an InGaAlAs quantum well and the light is guided in

a waveguide underneath [79]. The design details and applications of such optoelectronic device as a wide-band and

ultracompact optical modulator is discussed elsewhere [79]. It has been shown that for the system of interest, the quantum

well could be modeled as a three-level light emitting system with defined energy levels, corresponding to electrons, and

heavy and light holes. Transition dipole moments between different energy levels are also calcualted.

Table 1. Rabi frequencies.

(

) (

)

8.2640×1012 4.8067×1012 0.0068 0.0040

8.2640×1013 4.8067×1013 0.0681 0.0396

8.2640×1014 4.8067×1014 0.6815 0.3964

3.A.1. System characteristics

The light emitting system in [79] is a hetersotructure ( ) ( ) quantum

potential well. For an Aluminum fraction of x=0.9 and well material thickness of 9nm, the transition energy between

conduction and heavy hole bands will be about 0.8eV. There is a 30meV offset between the heavy and light holes bands.

We thus choose the energy levels in the modeled three level light emitter respectively as 0, 30, and 829 meV for light holes,

heavy holes, and electrons. Rabi frequencies are calculated as

( ) ⟨ | | ⟩ (33)

( ) ⟨ | | ⟩ (34)

in which and are the Rabi frequencies for transition between conduction band to heavy holes and light holes,

respectively. ⟨ | | ⟩ and ⟨ | | ⟩ are transition dipole moments, respectively calculated as 26.15 Debye and 15.21

Debye [79], is the applied electrical field, and is reduced Plank constant.

Table 1 shows the range of Rabi frequencies based on (33,34) for various electric field strengths. It was assumed that

the light emitting system is in intraction with photons having the wavelength of so the optical frequency would

be equal to

.

According to the Table 1. and because of near-resonance situation between light and electron-heavy hole transition, in

the weakest applied electric field with and , the system is in the weak coupling regime. For a

stronger electric field with and , the coupling regime is strong. Finally, for the strongest

electric field with and , the coupling enters the ultrastrong regime.

The general time-dependent state of the system and its describing Hamiltonian are

| ( )⟩ ∑ ∑ ( )| ⟩| ⟩ (35)

∑

∑ (

) .

/

(36)

In this system, is equal to zero because there is only one emitter. The coherent initial state is expressed as

| ( )⟩ √

∑ |√

|

∑ √

| ⟩ (| ⟩ | ⟩ | ⟩) (37)

7

3.A.2. Presence probabilities with Fock initial state

By applying Fock initial conditions we intended to study two cases. Firstly, to study the effect of boosting coupling

coefficient on the presence probability of the system in normalized time. Secondly, to study the importance of coefficient

matrix of the system without RWA. As we know, in RWA the effect of two terms ( )

and are neglected.

Since the light emitting system under consideration consists of three energy levels, so the effect of both and

could be studied. To study the

term, by the Fock initial condition, | ⟩ state is considered. According to (6),

under the operation of the state ket will become

| ( )⟩

)∑ ( ) | ⟩| ⟩ ∑ (

√ ∑ ( )| ⟩| ⟩ * +

| ⟩

√

| ⟩ √

| ⟩ (38)

Since, heavy to light hole transition is forbidden, we have

, and hence (38) becomes

| ( )⟩ √

| ⟩ (39)

Now the presence probability of system in the states | ⟩ | ⟩ are calculated and plotted as a function of normalized time

in Figs. 1-2.

To study the term due to Fock initial conditions, the state | ⟩ is considered. According to (6), under the

operation of the state ket becomes

Fig 1. The presence probability of the system in | ⟩ and | ⟩ ket states in upper and below rows, under weak, strong and coupling

regimes from left to right.

Fig 2. The presence probability of the system in | ⟩ and | ⟩ ket states under ultrastrong coupling regimes frome left to right.

Fig 3. The presence probability of the system in ket state of | ⟩, under weak, strong coupling regimes from left to right.

Fig 4. The presence probability of the system in ket state of | ⟩, under ultrastrong coupling regime and in ket states of | ⟩ and | ⟩

simultaneously, from left to right.

| ( )⟩ ∑ (

√ ∑ ( )| ⟩| ⟩ * +

| ⟩ √

| ⟩

(40)

Now by the considered fock initial conditions and (40), the presence probability of system in the ket state of | ⟩ | ⟩ is

measured and plotted as a function of normalized time in Figs. 3-4.

As it is visible from the above results, by the increase of coupling constant the frequency of collapse and revival events

is increased, except for the ultrastrong regime which exhibits a non-sinusoidal and disordered behavior. It is also observed

that although the probability of the states which are neglected in RWA is low in weak and strong coupling, it becomes

greater in the ultrastrong coupling. Hence, RWA is a very inappropriate approximation for study of ultrastrong coupling.

Fig. 5. Probablities of occupation of the heavy-hole (hh), light-hole (lh), and conduction (e) states in a, from left to right: weakly, strongly,

and ultrastrongly coupled system.

3.A.3. Presence probabilities with coherent initial state

Following (24), the presence probability of system at eah of the conduction, heavy hole, or light hole levels is

∑ ∑ | ( )|

(41)

This probablity is also found and plotted in the Fig. 5. By comparison it is observed that in the weak and strong coupling

regimes, the presence probability of the system in different states is sinusoidal on short time scales. This is while by

entering into ultracoupling regimes, the behavior is not sinusoidal at all on any time scale and is very chaotic both in short

and long temporal range.

3.A.4. Annihilation in different coupling regimes

Following (25), the expectation value of the field annihilation operator of the system is

⟨ ( )| | ( )⟩ ∑ ∑ √ ( )

( ) (42)

This expression has been calculated. Since the annihilator is non-Hermitian, its expectation values is complex-valued.

Making a three-dimensional parametric plot having the corresponding real and imaginary parts as functions of normalized

time is very instructive in this case. This has been shown for various coupling regimes and shown in Figs. 6-8. Also the

phase of expectation value as a function of normalized time duration has been also plotted in separate diagarams therein.

9

Fig. 6. Three-dimensional plot of the real and imaginary values and phase plot of the expectation value of the field annihilation operator

in weakly coupled system.


in strongly coupled system.


in ultrastrongly coupled system.

It is observed that in the weak coupling regime the behavior of the expectation value of the field annihilation operator is

completely sinusoidal and it confirms that solving Schrödinger equation in Heisenberg space with RWA can be acceptable

in this regime. By increasing the coupling constant and entering into strong regime, as it is seen in Fig. 7, the behavior

keeps varying sinusoidally, but the amplitude of oscialltions gradually decrease with time. This is while the phase behaves

just similarly to the weak coupling regime. For ultrastrongly coupled system, as shown in Fig. 8, however, the expectation

value plot is remarkedly non-sinusoidal and chaotic. At the same time, the phase is significantly non-linear. We had also

noticed this particular behavior of the phase under ultrastrong coupling in our recent studies [75-78].

3.A.5. Expectation value of the ladder operator

According to (27), the expectation value of the atomic ladder operator for transitions between conduction and heavy and

light holes are

⟨ ( )| | ( )⟩

∑ ∑ ( ) ( )

⟨ ( )| | ( )⟩

∑ ∑ ( ) ( )

(43)

The above expressions have been calculated and plotted in Figs. 9-14 as parametric plots similar to those of annihilator

operator in the above.

Fig. 9. Three-dimensional plot of the real and imaginary values and phase plot of the expectation value of in weakly coupled systems

from left to right.

Fig. 10. Three-dimensional plot of the real and imaginary values and phase plot of the expectation value of in strongly coupled

systems from left to right.

Fig. 11. Three-dimensional plot of the real and imaginary values and phase plot of the expectation value of in ultrastrongly coupled


Fig. 12. Three-dimensional plot of the real and imaginary values and phase plot of the expectation value of in a weak coupled systems

from left to right.

Fig. 13. Three-dimensional plot of the real and imaginary values and phase plot of the expectation value of in a strong coupled


11

Fig. 14. Three-dimensional plot of the real and imaginary values and phase plot of the expectation value of in a strong coupled


Similarly, as the system enters the ultrastrong coupling regime, chaotic behavior starts to develop, which is clearly visible

both in the phase space and phase plot.

3.A.6. Entanglement

The expectation value of the commutator of atomic ladder operators is known for simple quantum systems to encompass

basic information regarding the degree of entanglement. For multipartite systems, however, a more complicated measure

such as concurrence should be computed. For the system under study, we calculate and plot both. The expectation of

commutator of the ladder operators is found from

⟨ ( )| 0 ( ) 1 | ( )⟩

∑ | ( )|

∑ | ( )|

⟨ ( )| 0 ( ) 1 | ( )⟩

∑ | ( )|

∑ | ( )|

(44)

which has ben calculated and plotted for various transitions under weak, strong, and ultrastrong coupling in Figs. 15-16,

respectively for electron-heavy hole, and electron-light hole transitions. Again the general trend is such that the strongly

coupled system exhibits fast and slow components multiplied together, while this differentiation of fast and slow

oscialltions in the ultrastrongly coupled systems is not possible.

By recycling the computer software provided in our privous research [78], which self-generates an internal subroutine

for exact calculation of concurrence, we were also able to compute graphs of concurrence under various coupling strengths,

ranging from weak to strong and ultrastrong regimes. This has been shown in Fig. 17.

Fig. 15. Expectation value of [ ( )

] in weakly, strongly, and ultrastrongly coupled systems from left to right.



It is here concluded through observations made on many multipartite quantum systems that the entanglement of in all

cases behaves also chaotic in the ultrastrong regime and exhibits lots of disordered oscillations and distortion.

Further details of calculations and plots aswellas program source codes are presented for this system and other

examples in the thesis [80].

Fig. 17. Computed graphs of concurrence. From left to right: weakly, strongly, and ultrastrongly coupled systems.

3.B. CQED in a seven-partition system

In this section we report the simulation and analysis of a seven-partition system consisting of identical six quantum dots

interacting with one cavity mode. Due to the larger number of system partitions, the number of different states of the

system increases very much.

3.B.1. System specifications

The six quantum dots are here limited to two ground and excited states each, with energy eigenvalues of 1eV and 10eV,

respectively. The condition on identicality of dots may not be achieved in practive, and the developed software code is able

to treat different emitters with equal efficiency. The condition is set here for simplification of the problem and reduction of

the too many degrees of freedom. It is furthermore supposed that transition dipole moment in these quantum dots is 192

Debye. We also allow mutual dipole-dipole interactions between all the dots with a magnitude of 5meV. It is also assumed

that quantum dots are interacting with one resonant cavity mode, having the frequency

(45)

By applying different electrical fields and comparing with the optical frequency , different coupling regimes are

simulated. According to (33,34), Rabi frequencies as the coupling constants are here calculated and enlisted in Table 2, for

weakly, strongly, and ultrastrongly coupled systems.

Table 2. Rabi frequencies.

(

)

6.0676×1012 0.004

6.0676×1013 0.04

6.0676×1014 0.4

13

Here, for system is in weak coupling regime, for in the strong coupling, and for , the

coupling is ultrastrong.

With the assumption that the maximum possible number of photons in the cavity mode is eight, the general time

dependent state of system is expanded as

| ( )⟩ ∑ ∑ ( )| ⟩| ⟩

(46)

,…, denote the dots. The Hamiltonian with all parameters as introduced in the previous section will be

∑

∑ (

) .

/ ∑ (

)(

)

∑ (

)(

) (47)

Initial Fock and coherent conditions were considered to simulate the system. Based on (22) the coherent initial condition of

the system is:

| ( )⟩ √

∑ |√

|

∑ √

| ⟩

∑ | ⟩ (48)

3.B.2. Presence probabilities with Fock initial state

We first assume that the initial state is simply | ⟩, which expresses that there is exactly one photon in the

cavity mode and all quantum dots are in their ground energy level. Setting this ket as the initial state, we calculate and

plot the presence probability in this state as a function of normalized time in differnet coupling regimes in Fig. 18.

As it is seen in these plots, by increasing thecoupling constant the frequency of collapse and revival events also

increases, while in the ultrastrong regime the behavior is chaotic and not sinusoidal at all. This characteristic behavior of

the ultrastrong coupling, is also justified similarly in the rest of simulations, as discussed in the following.

Fig 18. The presence probability of the system in | ⟩ ket state under weak, strong and ultrastrong coupling regimes from left

to right respectively.

3.B.3. Presence probability with coherent initial state

The presence probability of a dot being in its ground, or excited energy levels is given according to (24) by

∑ ∑ | ( )|

(48)

∑ ∑ | ( )|

Fig. 19. Probablities of occupation of the excited and ground energy level states in the sixth quantum dot from left to right: simultaneously

in weakly, strongly coupled systems.

Fig. 20. Probablities of occupation of the excited and ground energy level states in the sixth quantum dot in ultrastrongly coupled system.

These probabilities for the sixth quantum dot have been plotted in the Fig. 19 for weak and strong, and in Fig. 20 for

ultrastrong coupling. The characteristic chaotic behavior of ultrastrong coupling can be seen again in Fig. 20.

3.B.4. Annihilation in different coupling regimes

The expectation value of the field annihilation operator of the system is given based on (25) as

⟨ ( )| | ( )⟩ ∑ ∑ √ ( )

( )

(49)

Phase space and phase plots of the real and imaginary values of the expectation value as functions of normalized time are

shown in Figs. 21-23, respectively for weakly, strongly, and ultrastrongly coupled systems.


in weakly coupled system.


in strongly coupled system.


in ultrastrongly coupled system.

It is surprisingly observed that the ultrastrongly coupled system not only exhibits a very chaotic and disordered behavior

in the three-dimensional parametric plot, but also the correspondin phase changes abruptly. This behavior is also seen in

15

nearly all other complex expectation values of all ultrastrongly coupled multipartite systems we have studied so far, and is

yet to be understood.

3.B.5. Expectation value of the atomic ladder operator

Again, we choose the sixth dot as the illustrative example. According to (27), the expectation value of the atomic ladder

operator for transition of every quantum dot individually from excited energy level to ground energy level is given by

⟨ ( )| | ( )⟩ ∑ ∑ ( )

( )

(50)

Three-dimensional (phase space) and phase plot of the real and imaginary values of this expectation value as functions of

normalized time duration was similarly plotted in Figs. 24-25.

Fig. 24. Three-dimensional plot of the real and imaginary values and phase plot of the expectation value of the sixth quantum dot in

weakly coupled system.


strongly coupled system.


strongly coupled system. As it is observed in numerical simulations, all six dots behave more or less according to the same pattern with slight

differences. So the same sinusoidal, or nearly-sinusoidal oscillations in weakly and strongly coupled systems, respectively,

are seen in the whole system. This is while the ultrastrong coupling is accompanied with chaotic three-dimensional

trajectories and multi-step random-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 and predicted correctly.



3.B.6. Entanglement

The expectation value of the commutator of atomic ladder operators for transition of every quantum dot individually from

excited energy level to ground energy level has been also analyzed following (32). Here, for the sixth quantum dot we have

⟨ ( )| 0 (

) 1 | ( )⟩ ∑ ∑ | ( )|

| ( )|

(51)

Plots are presented in Fig. 27.

It is observed and concluded by the measurements that the behavior of the entanglement of the system in all quantum

dots is chaotic in ultrastrong regime and has a lot of distortion.

4. CONCLUSION

In this paper, the general behavior of CQED of complex systems under different coupling regimes was analyzed.

Mathematically we tackled the most general quantum optical system consists of an arbitrary number of light emitters

interacting with an arbitrary number of cavity modes. We presented how to specify the general time dependent state of the

system, how to provide initial conditions and to solve the system without any approximation in time-domain in

Schrödinger picture. Next, we presented expressions for measuring presence probabilities, expectation value of field

operators, atomic operators, and commutators. We have developed an extensive MATLAB code to produce the necessary

initial conditions and solve the system. We also presented and discussed two example systems in details. We confirmed

that RWA may not be used in ultrastrong coupling. We furthermore have observed, for the first time, a chaotic behavior in

ultrastrong coupling regime accompanied by multi-step and random-like abrupt phase changes.

ACKNOWLEDGEMENTS

This work was supported in part by Iranian National Science Foundation under Grant 89001329.

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Cavity quantum electrodynamics of multipartite systems